Generation of iterated wreath products constructed from almost simple groups

Let G 1 G₁ , G 2 G₂ , … be a sequence of almost simple groups and construct a sequence ( W i ) (Wᵢ) of wreath products via W 1 = G 1 W₁=G₁ and, for each i > 1 i>1 , W i + 1 = G i + 1 wr W i Wᵢ₊₁=Gᵢ₊₁{wr}Wᵢ via the regular action of each G i Gᵢ . We determine the minimum number d ⁢ ( W i ) d(Wᵢ) of generators required for each wreath product in this sequence.