I have heard about the following result: for each finite simple non-abelian group $S$ and each natural number $r\ge 2$ there exists a number $n=n(r,S)$ such that the power $S^n$ is $r$-generator but $S^{n+1}$ is not $r$-generator. What is known about the numbers $n(r,S)$? Could someone give me references to this, please?

(I have posted this already on mathstackexchange.com, but did not get a response.)

Edit: This question is in a sense a converse to Bounding from below the cardinality of a set of generators of the $n$-fold cartesian product of a finite group. There it is basically asked for a given (arbitrary, finite) group $G$ and a given number $n$, how small can a generating set for $G^n$ possibly (not) be. In my question the input parameters were a finite simple group $S$ and a number $r\ge 2$ and the question was how big a number $n$ can possibly be so that $r$ elements are sufficient to generate the power $S^n$. Also I was interested in how this number (the biggest such $n$) is actually computed in concrete examples (or whatever is known about the computation of these numbers).

Basically I wanted to know, given a finite simple non-abelian group $S$ and a number $r$, the product of how many copies of $S$ do I need to take to get the $r$-generated free object in the formation generated by $S$.

@Editors/moderators: please feel free to delete the question if it is inappropriate.