In finite simple groups, most pairs of elements generate, so (at least asymptotically), the number of generating pairs is $\asymp |G|^2.$ See, for example:
Robert M. Guralnick, Martin W. Liebeck, Jan Saxl, and Aner Shalev, MR 1707675 Random generation of finite simple groups, J. Algebra 219 (1999), no. 1, 345--355.