So, to sum up (ha ha), I would say, that in speaking, "very many" in a negative phrase (such as: There aren't very many bees left.) is common and acceptable. But, I don't think I would use "very many" in a positive phrase, because in this case, it does seem redundant. We went to the countryside, where it was dark, and we saw very many stars in the sky. What? We can just say "many stars." Frankly, I would also never say "There are very many trees in a forest." This is not a judgement of anyone else's English, just an observation: that the phrase "very many" may be more common in some dialects of English than others. In writing, I agree with Randy. To make your writing crisper and snappier and to allow nouns to contain their full power, avoid using "very" if at all possible. For instance, if I write: That mineral is very rare. "Very" serves no purpose in the sentence and robs the word "rare" of its meaning. In other words, something is either rare or it is not. Take "very expensive," well, again, something is either expensive or it is not. Almost every usage of very is very similar. Ha. I mean similar. I even try to avoid writing "very good" if possible.

Your teacher who disapproves of "very" could be a writing teacher. While we accept "very" in spoken English, its use in writing does not represent preferred style. Trite comes to mind. "Very" adds nothing to meaning, and so in writing I recommend forgetting it.

"very many" is perfectly acceptable. One can also say "many many" but that's somewhat casual. "very many" can also be used in the negative as in "There aren't very many people who can speak five languages fluently." meaning "there are some but there aren't a lot"

I think "very many" is repetitive and I would avoid saying it. "Perfectably acceptable" and "absolutely fine" also might be considered extravagent uses of an adjective. What I am saying is: avoid using adjectives and adverbs unless you really need them.

You've had one answer from North America, one from South Africa, and here's one from England : yes, very many is absolutely fine. Native speakers from three continents agree with your teacher who says it's ok.