Finite generation of iterated wreath products

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Let (Gₙ,Xₙ) be a sequence of finite transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated permutational wreath product... G₂ G₁ is topologically finitely generated if and only if the profinite abelian group ∏ₙ≥ ₁ Gₙ/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... G 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].