Let $G$ be a finite nonabelian simple group. Must $G$ admit a pair of generators $g,h$ such that $g,[g,h]$ also generate?


I've computationally verified this for the first 21 nonabelian finite simple groups, though I don't know if this should hold in general.



Which nonabelian simple groups admit a generating pair $g,h$ such that $g,[g,h]$ is also generating?
Yes, this is true, at least for large enough groups. It is known that a random pair of elements generate a finite simple group (with probability approaching $1,$ as size goes to infinity), and also


*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.


Show that for random $x, y,$ the elements $x$ and $x^y$ generate with probability again approaching $1.$ as size goes to infinity.


The combination of the last two statements shows that your conjecture is true for large groups. It's quite possible that you can extract the statement for *all* simple groups by reading the referenced paper.


