  We investigate the production of three Higgs bosons at a proton-proton
collider running at a centre-of-mass energy of 100 TeV, all of which decay into
$b$-jets. This final state encapsulates by far the largest fraction of the
total cross section of triple Higgs boson production, approximately $20\%$. We
examine, by constructing detailed phenomenological analyses, two scenarios: (i)
one in which the triple and quartic Higgs boson self-couplings are modified
independently by new phenomena with respect to their Standard Model (SM) values
and (ii) an extension of the SM by a gauge-singlet scalar that could drive
first-order electroweak phase transition, within the context of the so-called
xSM. In the former, we find that competitive constraints of $\mathcal{O}(1)$
can be placed on the quartic coupling and in the latter we demonstrate that it
will be possible to obtain important information on the structure of the
extended scalar sector.
