Origins of the Chuquet Number Names
The following bit of text is from Nicolas Chuquet, Triparty en la Science des Nombres (1484) and presents the "powers of a million" number-naming system in its original form:
original words of Chuquet apparently transcribed
[...] pr[oc]eder. ¶ Item lon doit savoir que ung million vault mille milliers de unitez, et ung byllion vault mille milliers de millions, et [ung] tryllion vault mille milliers de byllions, et ung quadrillion vault mille milliers de tryllions et ainsi des aultres. Et de ce en est pose ung exemple : nombre divise et punctoye ainsi que devant est dit, tout lequel nombre monte 745324 tryllions 804300 byllions 700023 millions 654321. Exemple : 745324'8043000'700023'654321.
French: [...] to go. Item: one should know that a million is worth a thousand thousand units, and a byllion is worth a thousand thousand millions, and [a] tryllion is worth a thousand thousand byllions, and a quadrillion is worth a thousand thousand tryllions, and so on for the others. And an example of this follows, a number divided up and punctuated as previously described, the whole number being 745324 tryllions 804300 byllions 700023 millions 654321. Example : 745324'8043000'700023'654321.
As you can see, Chuquet intended the names to represent powers of 1000000 (the long scale or "billion=1012 system"). It is also clear that he was using prefixes that came from Latin, either directly or indirectly through French. To the more scientific mind it is more logical to use billion for 10000002, trillion for 10000003 and so on. In his example number there is an extra 0 that does not belong, 8043000 should be 804300. I suspect that the text pictured is a copy and that the error was introduced during transcription.
The immediately preceding line of text, not seen in the image, establishes Chuquet's use of number-names as high as nonillion and implied proposal to use Latin number prefixes for higher number-names "as far as you wish to go":
Ou qui veut le premier point peult signiffier million Le second point
byllion Le tiers poit tryllion Le quart quadrillion Le cinqe quyllion
Le sixe sixlion Le sept.e septyllion Le huyte ottyllion Le
neufe nonyllion et ainsi des ault's se plus oultre on vouloit pr[oc]eder.
French: Or if you prefer the first mark can signify million, the second mark byllion, the third mark tryllion, the fourth quadrillion, the fifth quyillion, the sixth sixlion, the seventh septyllion, the eighth ottyllion, the ninth nonyllion and so on with others as far as you wish to go. [...]
In 1514, Guillaume Budé (or Guilielmus Budaeus in Latin) introduced the word Milliart (milliard) for 1012, and the word spread with the help of Jacques Pelletier du Mans. (It was not used in the modern sense (109) until the late 1600's.)
Triparty en la Science des Nombres was copied and published by Estienne de La Roche in the 1520 textbook l'Arismetique, without attribution to Chuquet. 1520 is the date given by French dictionaries for the origin of the word billion9, but trillion is properly dated as coming from 1484.
Some History of Short vs. Long Scale
(A more thorough history is presented in the Wikipedia article Long and short scales.)
During the 17th century7 it became common practice in France to divide digits into groups of 3, and during this time billion began to be used as the name for 109 in France and Italy. French mathematicians decided to switch to this usage because they found it easier.2
From around that period came this quote, which has sometimes been placed alongside Chuquet's name or the above image, but is from a later writer:
Au lieu de dire mille milliers, on dira million, au lieu de
dire mille millions, on dira byllion, etc..., et tryllion,
quadrilion ... octylion, nonyllion, et ainsi des autres si plus
oultre on voulait proceder.
French: Instead of saying one thousand thousands, one may say million; instead of saying one thousand millions, one may say billion, etc..., and trillion, quadrillion, ... octillion, nonillion, and others as far beyond as you wish to go.
This passage reflects the short scale (or "billion=109") system. This meaning of the word billion came to the British American colonies (what would become the United States) in the early 1700's, and became common in France during the early 1800's (though France later switched back, in the mid 1900's).
The Chuquet number names were adopted throughout Europe (with minor spelling changes for each language), and used for both systems causing ambiguity when texts were translated. The billion=1012 system was adopted in England, Germany, Spain, Scandanavia, and eastern Europe except Russia8. Throughout the 1800's the French usage of the billion=109 system gained influence in the United States. As the world became larger and nations more interdependent, the ambiguity became an ever-greater problem, particularly when large amounts of money were being discussed.
Chuquet's manuscript was discovered by Aristide Marre in the late 1870s and published in 18802,6. In France, the 1948 General Conference on Weights and Measures deprecated the billion=109-system senses of the words billion through sextillion in favor of the billion=1012 system and this suggestion became official in 1961 throughout the French-speaking world (septillion and the higher number names were never part of the French language). France had finally2 switched back to the original version of the number names she had created. But the influence of the billion=109 system, primarily from the United States, was so great that by 1974, the Prime Minister of the U.K. announced that the billion=109 system was to be used for all official communications, effectively completing a transition that had long been taking place among the general population in English-speaking countries.
Latin Number Names
This table of Latin number names is adapted from several sources (16, 26, 36, 37, 38, 39). They can be used to formulate a system of number names extending on the Chuquet names. It will be useful as a reference when considering the proposals of writers throughout the 1800's and 1900's. Of these, the most consistent and well-researched is the Conway-Wechsler system from  (1995).
|1||I||unus, una, unum||primus|
|2||II||duo, duae, duo||secundus ; alter|
|3||III||tres, tres, tria||tertius|
|14||XIV||quattuordecim||quartadecimus ; quartus decimus|
|15||XV||quindecim||quintadecimus ; quintus decimus|
|21||XXI||viginti unus ; unus et viginti||unvicesimus ; vicesimus primus|
|22||XXII||viginti duo ; duo et viginti||duovicesimus ; vicesimus alter ; vicesimus secundus|
|23||XXIII||viginti tres ; tres et viginti||trevicesimus ; vicesimus tertius|
|24||XXIV|| viginti quattuor ; |
(etc.: there is a similar
[units] et [tens] form for
each number below 100)
|quartavicesimus (?) ; vicesimus quartus|
|25||XXV||viginti quinque||vicesimus quintus|
|30||XXX||triginta||trice(n)simus ; trigesimus|
|98||XCVIII||nonaginta octo ; duodecentum 26||duodecentesimus|
|99||XCIX||nonaginta novem ; undecentum 26||undecentesimus|
|122||CXXII||centum viginti duo||duovicesicentesimus (?) ; centesimus vicesimus secundus|
|200||CC||ducenti, -ae, -a||ducentesimus|
|300||CCC||trecenti, -ae, -a||trecentesimus|
|400||CCCC||quadringenti, -ae, -a||quadringentesimus|
|500||D||quingenti, -ae, -a||quingentesimus|
|600||DC||sescenti, -ae, -a||sescentesimus|
|700||DCC||septingenti, -ae, -a||septingentesimus|
|800||DCCC||octingenti, -ae, -a||octingentesimus|
|900||DCCCC||nongenti, -ae, -a||nonagentesimus|
|1124||MCXXIV||mille centum viginti quattuor|
|2000||MM||duo milia||bis millesimus|
|3200||MMMCC||tria milia ducenti|
|1,000,000||decies centena milia|
Chuquet's Dubious Use of Latin
To those with a reasonable working knowledge of Latin language, or anyone looking at the higher Latin numbers (as listed above) and comparing them later authors' extensions of the names beyond Chuquet's nonyllion, it becomes clear that Chuquet was not consistent in his use of Latin to form prefixes. One might also impugn the whole idea that Latin was the only source (perhaps it was French, or hallucination?)
Chuquet Names in the 1800s
In 1826, Dudley Leavitt published the book Pike's System of Arithmetick, beginning with a "Numeration" section describing how numbers are represented in print. The names millions, billions, trillions, etc. through duodecillions are used for powers of 1000000 (i.e. Leavitt used the "long scale") and always appeared in the plural, e.g. "Eight hundred millions three hundred forty-four thousand and two hundred". You can read the relevant text here: Dudley Leavitt, Pike's System of Arithmetick, 1826.
In 1856, Noble Heath published his A Treatise on Arithmetic, including a table of Latin number-names and examples of how to make these into names of powers of 1000 as high as centillion. You can read the relevant text here: Noble Heath, A Treatise on Arithmetic, 1856. This may well have been a personally-derived extension of the names up to duodecillion that were published in English at least as early as 1826 (Leavitt).
The earliest concerted effort I know of started with a "Professor Henkle" (most probably William Downs Henkle, from Ohio and born in 1828), whose ideas for number-names up to milli-millions=103000003 first appear in an 1860 edition of The Ohio Educational Monthly. He begins with a survey of earlier names (by Pike, Greenleaf, Loomis, Heath, Ray, Thomson, Holbrook) and discussion of the Latin derivation pointing out that the names are based on ordinal, not cardinal numbers (e.g. "quintillion" comes from quintus not quinque). You can read the full text of the article here: W. D. Henkle, Names of the Periods in Numeration, 1860.
Ad-hoc Chuquet Extensions
As described above, the commonly accepted number names like trillion come from Chuquet. It had already been established in Chuquet's time that prefixes based on Latin cardinal number-names would be used to express powers of a million, and the influence of Chuquet and Peletier was enough to cement an agreement to continue with that method as far as was wanted or needed.
As seen by comparing the names above with the powers of 1000 listed here, there are discrepancies even in a few of the names found in the highly-regarded print dictionaries: while tredecillion matches the Latin word tredecim, octodecillion departs completely from the Latin word for 18, which is duodeviginti. These discrepancies are similar to those Chuquet himself established with e.g. quintillion vs. Latin quinque=5 (omitting the que part of quinque).
Henkle points this out, in noting that "quintillion"/"sextillion"/"nonillion" clearly have far greater resemblance to the Latin ordinals quintus/sextus/nonus. (If they had been based on the cardinals quinque/sex/novem, we would have something like "quinquillion"/"sexillion"/"novillion".) Henkle therefore suggests using Latin cardinal syllables throughout; but others before and after Henkle had difficulty, as the numbers get higher the Latin ordinals become less definitive and more awkward.
Things get a little worse when going beyond vigintillion because there is no good Latin prefix for twenty-one. Just as in English, starting with 21 the name for the number goes to two words, with the smaller part as the second word. Those who have extended Chuquet names beyond vigintillion usually go to something like unvigintillion, breaking away from the Latin but keeping the similarity to undecillion. This is what Conway and Wechsler do. For example, in the short scale, the Conway and Wechsler name for 1090 is novemvigintillion, but I have also seen vigintinonillion apparently based on some combination of the cardinal viginti "twenty" and ordinal nonus "ninth"23 despite that undetriginta and undetricesimus were by far the most common way to say "29" and "29th" in Latin.
The question of whether to use elision (omitting letters for more natural pronunciation) results in the distinction between sexdecillion and sedecillion for 1051, tresvigintillion and trevigintillion for 1072, and so on.
There seems to be a cult-like appeal to the Chuquet number names, evident in the fact that many people have been inspired to create systems, and none seem to want to use each other's system. In this regard the Chuquet extension phenomenon resembles the cult of 137 that has built up around the Sommerfeld fine-structure constant. What follows is a partial survey of personal, "ad-hoc" Chuquet-like naming systems that have been created over the years, with relevant links.
Henkle's names were re-published in 1876 by Brooks then again in 1904 . These were listed in 1968 by Borgmann  in the premiere issue of his Word Ways: the Journal of Recreational Linguistics. By that point it seems no-one knew who "Henkle" was , though it should have still been apparent that his knowledge of the Latin language was adequate for the task. It is also notable that the original article by W. D. Henkle gives the name milli-millions to the 1000000th power of 1000000; only with the 1876 reprinting by Brooks did the missing syllable get restore to make "milli-millillions".
The Borgmann article sparked several follow-on articles in Word Ways presenting Chuquet-like extension proposals, notably Ondrejka in 1968  (who proposed a system that goes up to milli-millimillillion=103000000003); and Candelaria in 1975, 1976 and 1983 , (who redefined milli-millimillillion to be 103×103000000+3 and topped out at nona-centillillesillillion=103×102700000000+3.
In the fall 1980 presentation of Stanford's CS 204, Donald Knuth and Allen Miller presented some problems relating to alphabetical ordering of numbers.30 In this they defined 10213 to be septagintillion rather than the more commonly-seen septuagintillion; similarly 10243 was octagintillion rather than octogintillion.
John Knoderer has created a set of number names between vigintillion and centillion, differing somewhat from the above. For example, he gives septoctogintillion for 10264, whereas the Conway-Wechsler name is septemoctogintillion His names are listed on his Numbering Systems & Place Values page. A similar list is provided by Sally Berriman.
Gregg William Geist23 applied Latin names more strictly than most others, even going so far as to adapt the standard Latin undeviginti=19 into the name undevigintillion for 1060, and similarly undetrigintillion for 1090, etc. He also puts the bigger part first for everything above vigintillion: vigintimillion=1066, vigintisextillion=1081, centumsedecillion=10351, etc.
A more original system based on Greek prefixes has been proposed by Russ Rowlett. His main stated goal is to avoid the ambiguity of the existing Latin-based names entirely by replacing them all. He starts with gillion for 109 inspired by the SI prefix giga-, then uses Greek number names as prefixes: tetrillion=1012, pentillion=1015, hexillion=1018, ... icosillion=1060, ..., triacontillion = 1090, and so on. The Greek number prefix corresponds to the power of 1000, thus tetr- for 4, and tetrillion=10004.
Louis Epstein uses SI-like prefixes (e.g. kilillion=10000001000=106000, zetillion=106×1021) reminiscent of some of the smaller Bowers names for certain class-3 numbers, and also combines multiple Greek parts with Latin parts in elaborate ways. For example, he states "The 1048576th power of a million is a sexseptaginquinhectooctoquatriginkilmegillion"5; the hecto, kil and meg parts come from Greek while the other parts are the familiar Latin syllables used in most of the other systems.
These systems have plenty of differences, and frequently even ambiguity or lack of self-consistency. Going beyond 103003 (or 106000 if you choose the billion=1012 system a.k.a. "long scale") allows for even more variations, mainly from the use of a large number of prefixes strung together and having to remember what order they go in. Perhaps to address this issue, Landon Curt Noll supplements his extensive description, lists and tables with a page that automaticlly converts numbers to names, with downloadable source code for UNIX users like me. A few examples of his system: 1019683 is "one sexmilliaquingensexagintillion", 103000003 is "one milliamilliatillion", and 101010 = 1010000000000 is "ten tremilliamilliamilliatrecentretriginmilliamilliatrecentretriginmilliatrecendotrigintillion". His system has a name for googolplex that is two words long, "ten tremilliamilliamillia...milliatrecentretriginmilliatrecendotrigintillion"; the second word has 3903 letters.
More Conway-Wechsler Number Names
The following table shows examples of extensions to the Chuquet names using the most consistent and well-researched extension (that by Conway and Wechsler in , including the Miakinen suggestion regarding quin-). The rightmost column shows how some others have dealt with the task of turning the Latin into a prefix.
|N||N in Latin 3,17||103N+3||Conway-Wechsler name  for 103N+3||other names|
|16||sedecim ; sexdecim||1051||sedecillion||sexdecillion18,30|
|21||viginti unus ; unus et viginti||1066||unvigintillion||primo-vigillion25|
|22||viginti duo ; duo et viginti||1069||duovigintillion||dovigintillion24|
|25||viginti quinque ; etc.||1078||quinvigintillion||quinquavigintillion19|
|99||nonaginta novem ; undecentum 26||10300||novenonagintillion|
|121||centum viginta unus||10366||unviginticentillion||primo-vigesimo-centillion25|
|133||centum triginta tres||10402||trestrigintacentillion|
|333||trecenti triginta tres||101002||trestrigintatrecentillion|
|1234||mille ducenti triginta quattuor||103705||milliquattuortrigintaducentillion|
|6560||sexies milia quingenti sexaginta||1019683||sextillisexagintaquingentillion|
|19683||undevicies milia sescenti octoginta tres||1059052||novendecillitresoctogintasescentillion|
|1000000||decies centena milia||103000003||millinillinillion||milli-millillion25|
The Conway-Wechsler system disagrees with some dictionaries regarding quinquadecillion, sedecillion, and novendecillion (which are better known as quindecillion, sexdecillion and novemdecillion respectively)18,20,21. Miakinen22 explains that sedecillion and novendecillion are more true to the "rules of assimilation" in Latin, and thus the Conway-Wechsler version is better. But he also explains that quinquadecillion should be quindecillion because the Latin for 15 is "quindecim", not "quinquadecim", and proposes a similar change to all the Conway-Wechsler names involving the quinqua- prefix; I have adopted his suggestion here.
Notably, they are identical to the Henkle names for many values of 10300N+3 up to millillion.
Beyond 103000, the Conway-Wechsler system combines multiple parts of the names for smaller numbers to form a name for the larger number. The rules are described here. In the table above, you can see several more examples. A convenient pattern emerges: 103006 is one millimillion, which combines an extra milli- with the familiar name million for 106. Similarly, 103009 is millibillion, 103000 times a billion, and so on.
This pattern continues all the way up: compare the names above for 101683 and 1019683: they differ only by the addition of an initial sextilli- which is derived from sextillion. sextillion is 1021, and sextilli- is found at the beginnings of Conway-Wechsler names up to 1021000 (and starting with 1018003).
Successful use of the Conway-Wechsler system depends on careful attention to the details of the assimilation rules for combining multiple prefixes. For example, someone could conclude that 10312 trescentillion and 10903 trecentillion had the same name. However, if the rules are followed completely (as described here) there are no spelling ambiguities. (Pronunciation, however, is still an issue!)
Also of note is Wechsler's comment:
The presentation in The Book of Numbers was designed to leave the impression that the system up to 999 was pre-existing, although in fact we invented a lot of it; you will note that Conway carefully never says the system is ancient. -- Allan Wechsler21
Although Conway's book does state (emphasis added):
You can now use the usual rules for combining this complete system of zillion words (which first appears in the present Book of Numbers) so as to obtain correct 'English names', like [...] -- Conway and Guy 
Perhaps The Book of Numbers will trip up one or two future etymology researchers, but the inconsistency with quindecillion, sexdecillion and novemdecillion should make the truth obvious.