  In 1985, Odoni showed that in characteristic $0$ the Galois group of the
$n$-th iterate of the generic polynomial with degree $d$ is as large as
possible. That is, he showed that this Galois group is the $n$-th wreath power
of the symmetric group $S_d$. We generalize this result to positive
characteristic, as well as to the generic rational function. These results can
be applied to prove certain density results in number theory, two of which are
presented here. This work was partially completed by the late R.W.K. Odoni in
an unpublished paper.
