Øystein Ore proved in 1951 that for $n\gt4$ we have that all elements of $A_n$ are commutators.  He subsequently conjectured that this is true for all finite non-abelian simple groups.

In this connection,  not long ago I asked [if every element of a perfect group is a commutator][1]. It was answered in the negative. I might have caught that $A_5*A_5$ is a counterexample.   The smallest such was found to be of order 960.

Ore's conjecture/theorem was proved in 2008, and relies on the *classification theorem*.  

 - See M. W. Libeck, E. A. O'Brien, 
Aner Shalev, Pham Huu Tiep,  *The Ore conjecture*, Journal of the European Mathematical Society, vol. 12, issue 4, 2010.  pp. 939-1008. http://doi.org/10.4171/JEMS/220

Interestingly Ore himself was mainly concerned with graph theory. 


  [1]: https://math.stackexchange.com/q/4865510/1070376