  For any finite group $A$ and any finitely generated group $B$, we prove that
the corresponding lamplighter group $A\wr B$ admits a standard generating set
with unbounded depth, and that if $B$ is abelian then the above is true for
every standard generating set. This generalizes the case where $B=\mathbb{Z}$
together with its cyclic generator due to Cleary and Taback. When $B=H*K$ is
the free product of two finite groups $H$ and $K$, we characterize which
standard generators of the associated lamplighter group have unbounded depth in
terms of a geometrical constant related to the Cayley graphs of $H$ and $K$. In
particular, we find differences with the one-dimensional case: the lamplighter
group over the free product of two sufficiently large finite cyclic groups has
uniformly bounded depth with respect to some standard generating set.
