  At sufficiently high energies the production of a very large number of
particles is kinematically allowed. However, it is well-known that already in
the simplest case of a weakly-coupled massive $\lambda \varphi^4$ theory,
$n$-particle amplitudes become non-perturbative in the limit where $n$ scales
with energy. In this case, the effective expansion parameter, $\lambda n$, is
no longer small and the perturbative approach breaks down. In general, the
associated $n$-particle production rates were argued to be described by an
exponential that, depending on the specifics of the underlying Quantum Field
Theory model, could be either growing or decaying in the large-$n$ regime. We
investigate such processes in general settings of Effective Field Theory (EFT),
involving arbitrary higher-dimensional operators of $\varphi$. We perform the
resummation of all leading loop corrections arising from EFT vertices for
amplitudes at the multiparticle threshold. We find that the net effect of
higher-dimensional operators amounts to an exponentially growing factor. We
show that if an exponential growth was already generated by the renormalizable
interactions, it would then be further enhanced by the EFT contributions. On
the other hand, if the multiparticle rates computed in the renormalizable part
of the theory were suppressed, this suppression would not be lifted in the EFT.
