Some History of Short vs. Long Scale Names
During the 17th century7 it became common practice 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 The misquote above might be from around this time.
Extending this pattern to the other Chuquet names (with trillion a thousand times larger than billion, and so on) created the short scale system. For the rest of this section I will call this the "billion=109 system".
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. During the 19th century 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 speaking population.
Latin Number Names
This table of Latin number names is adapted from an online Latin textbook16. They can be used to formulate a system of number names extending on the Chuquet names. The most consistent and well-researched such system is the Conway-Wechsler System from .
|1||I||unus, una, unum|
|2||II||duo, duae, duo|
|3||III||tres, tres, tria|
|21||XXI||viginti unus ; unus et viginti|
|22||XXII||viginti duo ; duo et viginti|
|23||XXIII||viginti tres ; tres et viginti|
|24||XXIV|| viginti quattuor ; |
(etc.: there is a similar
[units] et [tens] form for
each number below 100)
|90||XCIX||nonaginta novem ; undecentum 26|
|122||CXXII||centum viginti duo|
|200||CC||ducenti, ae, a|
|300||CCC||trecenti, ae, a|
|400||CCCC||quadringenti, ae, a|
|500||D||quingenti, ae, a|
|600||DC||sescenti, ae, a|
|700||DCC||septingenti, ae, a|
|800||DCCC||octingenti, ae, a|
|900||DCCCC||nongenti, ae, a|
|1124||MCXXIV||mille centum viginti quattuor|
|3200||MMMCC||tria milia ducenti|
|1,000,000||decies centena milia|
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.
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.
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).
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 based on the Latin viginti novem23.
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.
I myself extended the Chuquet names at age 10, inventing names like bigintillion=1093, cigintillion=10123, etc. (continuing with different letters of the alphabet) to permit successive sets of ten names like unbigintillion=1096, duobigintillion=1099, and so on. This system had plenty of problems for example, in a hypothetical standard American English pronunciation, at least two of cigintillion, kigintillion and sigintillion must sound the same. Soon I realized that I could have gotten more range by using dipthongs (blingintillion, bringintillion, ...) and that my system was redundant to the existing standard my bigintillion is the same as the standard trigintillion, etc. Within a month or two I switched to scientific notation and never looked back.
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-millillion=103000003 first appear in an 1860 edition of The Ohio Educational Monthly (another version here). 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).
Henkle's names were re-published in 1904 by Brooks . 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. The Henkle names differ in most cases but are notably identical to the Conway-Wechsler names for many values of 10300N+3 up to millillion.
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 and 1976 , (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.