This depends on the context. If you just mean something to write on, "paper" is uncountable. But "paper" can also mean things that are countable: newspapers, essays for school, and other kinds of documents.
So, for example:
1. (in class)
A: I forgot my notebook. Do you have any paper I can borrow?
B: Yes, I have plenty of PAPER.
2. (at a newsstand)
A: Do you have any newspapers?
B: Yes, I have plenty of PAPERS.
3. (in university)
A: Do you have to write a lot of essays this week?
B: Yes, I have plenty of PAPERS.