Part of speech of "very," "extremely," "really," and "quite"

One problem is that the entire concept of "part of speech" is very old. How we use it in English, especially in dictionaries, goes back to the study of Latin and Greek. In this view of English grammar "adverb" is the catch-all category where everything that doesn't fit into one of the other traditional categories ends up. (The others being noun, verb, adjective, pronoun, preposition, conjunction, and interjection.)

Now there is no one, true description of any language (except perhaps constructed languages such as yours). There are merely alternative or competing descriptions which appear over time as more independent analyses of the language are undertaken. Such descriptions or analyses may be called "grammars".

Most (but not all) grammars include a concept of word class under one name or another. So one problem is that "part of speech" has two meanings. One is the specific set of eight categories from the classical languages, the other is as a synonym for word class, which is a lot looser.

So all your example words are adverbs under this older stricter view of parts-of-speech, but their qualities and quirks can be much more thoroughly investigated in newer ways. And various new ways will have various new terms for the classes they put these various words into.

Unless you are inventing a new language specifically to embrace the classical parts of speech you don't have to worry in which they belong, but if you are inventing a new language to learn more about how language works then it will be worth your time reading up on the many newer grammars and language descriptions and analyses.

In some contexts So and too are a couple more of these kind of words, but there probably aren't that many altogether.

By current definitions I think they're just adverbs (as 'broadly' defined here). If you want finer granulation in the terminology you can say they're adverbs that modify adverbs, or invent your own term. Or have one on me - adadverbs or metadverbs, take your pick.

I also agree with Alenanno that they're sub-modifiers. Not because I ever came across that usage before, but it seems logical enough. On the other hand, I imagine that linguistic category is somewhat larger than the one we're focussing on.

LATER - thefreelibrary says you can call them intensifying adverbs, intensifiers, or intensives. Also throws in a few more examples: absolutely, especially, precisely.