  It has been suggested that scattering cross sections at very high energies
for producing large numbers of Higgs particles may exhibit factorial growth,
and that curing this growth might be relevant to other questions in the
Standard Model. We point out, first, that the question is inherently
non-perturbative; low orders in the formal perturbative expansion do not give a
good approximation to the scattering amplitude for sufficiently large N for any
fixed, small value of the coupling. Focusing on $\lambda \phi^{4}$ theory, we
argue that there may be a systematic approximation scheme for processes where N
particles near threshold scatter to produce N particles, and discuss the
leading contributions to the scattering amplitude and cross sections in this
limit. Scattering amplitudes do not grow as rapidly as in perturbation theory.
Additionally, partial and total cross sections do not show factorial growth. In
the case of cross sections for $2 \to N$ particles, there is no systematic
large N approximation available. That said, we provide evidence that
non-perturbatively, there is no factorial growth in partial or total cross
sections.
