  Let $(G_n,X_n)$ be a sequence of finite transitive permutation groups with
uniformly bounded number of generators. We prove that the infinitely iterated
permutational wreath product $...\wr G_2\wr G_1$ is topologically finitely
generated if and only if the profinite abelian group $\prod_{n\geq 1} G_n/G'_n$
is topologically finitely generated. As a corollary, for a finite transitive
group $G$ the minimal number of generators of the wreath power $G\wr...\wr G\wr
G$ ($n$ times) is bounded if $G$ is perfect, and grows linearly if $G$ is
non-perfect. As a by-product we construct a finitely generated branch group,
which has maximal subgroups of infinite index, answering [2,Question 14].
