  We investigate composition operators on Hardy-Orlicz spaces when the Orlicz
function $\Psi$ grows rapidly: compactness, weak compactness, to be
$p$-summing, order bounded,..., and show how these notions behave according to
the growth of $\Psi$. We introduce an adapted version of Carleson measure. We
construct various examples showing that our results are essentially sharp. In
the last part, we study the case of Bergman-Orlicz spaces.
