  Let $\mathcal{S}$ be a sequence of finite perfect transitive permutation
groups with uniformly bounded number of generators. We prove that the
infinitely iterated wreath product in product action of the groups in
$\mathcal{S}$ is topologically finitely generated, provided that the actions of
the groups in $\mathcal{S}$ are not regular. We prove that our bound has the
right asymptotic behaviour. We also deduce that other infinitely iterated mixed
wreath products of groups in $\mathcal{S}$ are finitely generated. Finally we
apply our methods to find explicitly two generators of infinitely iterated
wreath products in product action of special sequences $\mathcal{S}$.
