A Godzillion, named after Godzilla, is a "number" allegedly invented by Canadian author, humorist, and radio producer Jonathan Goldstein, which became publicly known through his radio show WireTap on CBC Radio One.
It is defined as "The largest number ever. The monster of numbers. A number exceeding all other numbers." according to Urban Dictionary.
However, earlier references to this number that define it in a similar way exist, such as in the Brag of the SubGenius, which says in part, "The say a godzillion is the highest number there is. Well by God! I count to a godzillion and one!"
Since the Brag of the SubGenius was written in 1980 by Douglass St. Clair Smith (a.k.a. Ivan Stang), and Jonathan Goldstein first mentioned this number in the 2000s, it is likely that the "godzillion" neologism was one of the many neologisms coined by Ivan Stang, who has invented many new words, most of them having nothing to do with math, published in pamphlets and books associated with the Church of the SubGenius, a parody religion he co-founded in 1979.
The most likely origin of this word is from the Brag of the SubGenius, although it is possible that it may have been used even earlier and Ivan Stang may have heard of it somewhere else. It is unknown whether Jonathan Goldstein independently came up with this "number", heard of it from a SubGenius friend of his, or if he actually invented it prior to Stang and it was Stang who heard of it from Goldstein.
Given that Goldstein's work first became publicly known around the year 2000 and Stang's work first became publicly known in 1979, the most likely explanation is that Ivan Stang invented this joke "number" and Jonathan Goldstein heard about it somewhere and mentioned it on the radio, and some people mistakenly attributed it to Goldstein instead of Stang.
However, the true origins of this word are not entirely known for sure. It does seem very likely, though, that the origin of this word is a combination of "Godzilla" with "gazillion", which means, this word was definitely coined some time after the first Godzilla movie and after gazillion became a slang term, meaning, this word was clearly coined some time after 1954 when the first Godzilla movie was made, and definitely existed by 1980 when it appeared in the Brag of the SubGenius pamphlet written by Ivan Stang, and even more so definitely existed when Jonathan Goldstein discussed it on CBC radio in the 2000s.
Since all popular definitions of this word rely on it being the biggest number, bigger than all other numbers, a godzillion would definitely be the largest of the ill-defined joke "numbers", by definition. Whether it is bigger or smaller than infinity depends on whether or not you consider infinity a number. Of course, since you can always just do something like add one to a godzillion to get an even bigger number, its definition of being the largest number is self-contradictory, which is, of course, what makes this number a joke and not something anyone in the googology community or any other mathematician would take seriously. If you do consider infinity to be a number, though, it is possible that, rather than being bigger than infinity, a godzillion would simply be exactly the same thing as infinity, just a new word for that existing concept... the same trick of adding one can also be done with infinity but infinity is still regarded as a valid concept.