  The syzygies of the d-th Veronese embedding of $\mathbb P(V)$ are functors of
the complex vector space V. From a certain perspective, we show that as d
grows, their Schur functor decomposition is very rich whenever they are not
zero. This is deduced from an asymptotic study of related plethysms. We also
obtain other results related to a question of Ein and Lazarsfeld.
