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.