P. Hall gave a formula for the number of generators of $G^n$ for any finite simple group $G$. One famous example is the fact that $A_5^{19}$ is 2-generated, but $A_5^{20}$ is not. The question of computing $d(G^n)$ has extensively been studied, starting with the work of Wiegold. However, what we need is a bound for the direct product of different factors. More precisely, for $k_1<k_2<\dots<k_r$, how large can we take the exponents $e_i$, such that $A_{k_1}^{e_1}\times\dots\times A_{k_r}^{e_r}$ is still 2-generated?

The closest related result I could find is due to Kravchenko and Petrenko (Some formulas for the smallest number of generators for finite direct sums of matrix algebras, Theorem 2.6), where the corresponding problem for full matrix algebra is dealt with. I would assume that the answer for simple groups is similar to the one for matrix algebras. The formula $d(A_{k_1}^{e_1}\times\dots\times A_{k_r}^{e_r})=\max_i d(A_{k_i}^{e_i})$ might be too optimistic, but I guess that it is not far from the truth.