I'm not an ESL teacher, but I'm a native speaker.
There are only a few cases in which I would consider "very great" as sounding correct.
- He was a very great man. (sounds fine to me)
- There are a very great number of people/issues. (sounds awkward to me)
A quick Google search reveals that "very great" is used occasionally in the titles of both recent and old books/articles:
- The Life we have is very great
- A Very Great Profession: The Woman's Novel
"Very great" also appears frequently in different translations of Bible verses. I think it may have had common usage some time ago. You're right, though. It's probably best not to use it.
On the other hand, "really great" (which should mean the same as "very great") sounds fine to me in many contexts.
- He's a really great guy.
- That was a really great book.
- They're doing some really great things.
Student or Learner