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:

<cite authors="Robert M. Guralnick, Martin W. Liebeck, Jan Saxl, and Aner Shalev" mrnumber="1707675" cite="_J. Algebra_ **219** (1999), no. 1, 345--355">_Robert M. Guralnick, Martin W. Liebeck, Jan Saxl, and Aner Shalev_, MR 1707675 [**Random generation of finite simple groups**](http://dx.doi.org/10.1006/jabr.1999.7869), _J. Algebra_ **219** (1999), no. 1, 345--355.</cite>