At very high energies scattering amplitudes in a spontaneously broken gauge theory into multiparticle final states are known to grow factorially with the number of particles produced. Using simple scalar field theory models with and without the vacuum expectation value, we compute total cross sections with up to seven particles in the final state at the leading order in perturbation theory with MadGraph. By exploring the known scaling properties of the multiparticle rates with the number of particles, we determine from these the general n-point cross sections in the large-n limit. In the high-multiplicity regime we are considering, n≫1 and λn=fixed, the perturbation theory becomes strongly coupled with the higher-order loop effects contributing increasing powers of λn. In the approximation where only the leading loop effects are included, we show that the corresponding perturbative cross sections grow exponentially and ultimately violate perturbative unitarity. This occurs at surprisingly low energy scales ∼40–50 TeV with multiplicities above ∼150. It is expected that a repair mechanism or an extension of the theory has to set-in before these scales are reached, possibly involving a novel nonperturbative dynamics in the a priori weakly coupled theory.