Is it true that every finitely generated infinite simple group has exponential (word-)growth?
Remark: As Mark Sapir has pointed out, the question whether every finitely generated group of subexponential growth is even residually finite has been answered in the negative in
Anna Erschler. Not residually finite groups of intermediate growth, commensurability and non-geometricity, J. Algebra 272 (2004), no. 1, 154–172, http://www.sciencedirect.com/science/article/pii/S0021869303006410.