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