Generation of iterated wreath products constructed from alternating, symmetric and cyclic groups

Let $G_{1}$, $G_{2}$, ... be a sequence of groups each of which is either an alternating group, a symmetric group or a cyclic group and construct a sequence $(W_{i})$ of wreath products via $W_{1} = G_{1}$ and, for each $i \geq 1$, $W_{i+1} = G_{i+1} \operatorname{wr} G_{i}$ via the natural permutation action. We determine the minimum number $d(W_{i})$ of generators required for each wreath product in this sequence.