Why does it work with "a lot", but not "many times"? I think "a lot" means "many times".
I agree with David and find his explanation perfectly convincing. I especially like this definition of his:
As to a reason, "a lot" is more of an intensifier than a quantifier.
"many times" expresses the repetition of an action (an action that starts and finishes, then starts and finishes again, and so on) and this does not fit in with the present perfect continuous which indicates, as the name of the tense goes, continuity.
If instead of "do an exercise" (or "do exercises") we used "practice" (which can be deemed to be a close synonym of "do exercises," for instance as preparation for a test), we could perhaps see even more clearly that "a lot" works but "many times" does not:
- I've been practicing a lot.
- I've been practicing many times.