Such a question had never occurred to me but if asked, my initial response would have been “A lot”. But I would have been wrong. It turns out that the number is surprisingly small and that I had (once again) been misled by the deceptive power of geometric progression.
I’ll let readers have the fun of guessing for themselves (assume that you can have a piece of paper of any size to start with) and then they can read this New Scientist report about a group of students who worked on this question for seven years before breaking the previous record.
It turns out that there is some fascinating physics involved in crumpling paper.