How did the comparative of "well" become "better"? At what point in history did this happen? The same with the superlative, "best'? Were there other, older forms that were dropped for some reason (weller, wellest)?

How did the comparative of "well" become "better"?

An interesting question. First, this is called suppletion (or suppletive forms) when a word has different forms that come from different roots. You see it in the paradigm for be (am, are, is, was, were, being, been) and in go (goes, went, going, going). Went was originally the past form of wend. I have never read anything on why suppletion happens, and like most of language change I think it would be nearly impossible to say, but we can often trace these changes.

Now to good and well. In Present-Day English, most adverbs are formed from adjectives by adding the suffix -ly. This morphological process stems from the Middle English periuod. In Old English adjective were regularly formed by adjectives by the addition of the suffix -e (fæger 'pretty, beautiful' (whence [i[]fair[/i]), fægre 'pleasantly, beautifully'). The suffix -lic was in Old English adjectival and there was a corresponding form in -e to make adverbs from adjectives, -lice. Final e soon became unstressed schwa and finally silent, so the two suffixes were sounded the same, and then got reanalyzed as purely an adverbial suffix. OE wel comes from the same root in Proto-Germanic as the auxiliary verb will.

Now part of the story is why we use adverbs to qualify verbal action, when most Germanic languages use plain old adjectives. In German is is grammatical to say schlaff gut 'sleep well'. Well is also an adjective in English: I am well. I hope that helps.

Since English thruout its history has had this thing where the comparative and superlative of good are supplied by better and best, and German has it too, I'd say it happened sometime before any Germanic languages were written down.

I'd say it happened sometime before any Germanic languages were written down.

Yes. In Latin, bonus 'good', melior 'better', optimus 'best' is suppletive, too. Not related. I do know that English better is related to the Sanskrit name Subhadra (with su- prefix (same as Greek eu-) meaning 'good'.

So what you say is

bonus, melior, optimus
good, better, best
gut, besser, best
goed, beter, best

are grabbed together form different roots?

melior 'better', ! Ameliorate! Eh?

different roots

Well, the better and best ones may be from the same root. But all the others are definitely from different ones.

melior 'better', ! Ameliorate! Eh?

Yep. And the bad paradigm in Latin is malus, pejor, pessimus.

good, gut and goed are all related, and all the Germanic words for better and best are related. better is thought to go back to Proto-Germanic *batiz, and best to the superlative form *batist.