  We carry out a detailed study of multi-Higgs production processes in the
gluon fusion channel in the high energy regime relevant to Future Circular
hadron colliders and in the high-Higgs-multiplicity limit (> 20). Our results
are based on the computation of the leading polygons - the triangles, boxes,
pentagons and hexagons - to the scattering processes, further combined with the
subsequent branchings to reach high final state multiplicities. The factorial
growth of the number of diagrams leads to an exponential enhancement of such
large multiplicity cross-sections and, ultimately, in breaking of
perturbativity. We find that the characteristic energy and multiplicity scales
where these perturbative rates become highly enhanced and grow with increasing
energy are within the 100 TeV regime with of the order of 130 Higgses (or more)
in the final state. We also show that already for a 50 TeV hadron collider the
perturbative cross-sections for 140 bosons are at picobarn level.
