the product replacement graph (prg) of a group g is the set of generating k-tuples of g, with edges corresponding to nielsen moves. it is conjectured that prgs of infinite groups are nonamenable. we verify that prgs have exponential growth when g has polynomial growth or exponential growth, and show that this also holds for a group of intermediate growth: the grigorchuk group. we also provide some sufficient conditions for nonamenability of the prg, which cover elementary amenable groups, linear groups, and hyperbolic groups.