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 $i\ge 1$, $W_{i+1}=G_{i+1}\mathrm{wr}{G}_i$ via the natural permutation action. We determine the minimum number $d(W_i)$ of generators required for each wreath product in this sequence.