  The protection number of a vertex $v$ in a tree is the length of the shortest
path from $v$ to any leaf contained in the maximal subtree where $v$ is the
root. In this paper, we determine the distribution of the maximum protection
number of a vertex in simply generated trees, thereby refining a recent result
of Devroye, Goh and Zhao. Two different cases can be observed: if the given
family of trees allows vertices of outdegree $1$, then the maximum protection
number is on average logarithmic in the tree size, with a discrete
double-exponential limiting distribution. If no such vertices are allowed, the
maximum protection number is doubly logarithmic in the tree size and
concentrated on at most two values. These results are obtained by studying the
singular behaviour of the generating functions of trees with bounded protection
number. While a general distributional result by Prodinger and Wagner can be
used in the first case, we prove a variant of that result in the second case.
