  The Higgs potential consists of an unexplored territory in which the
electroweak symmetry breaking is triggered, and it is moreover directly related
to the nature of the electroweak phase transition. Measuring the Higgs boson
cubic and quartic couplings, or getting equivalently information on the exact
shape of the Higgs potential, is therefore an essential task. However, direct
measurements beyond the cubic self-interaction of the Higgs boson consist of a
huge challenge, even for a future proton-proton collider expected to operate at
a center-of-mass energy of 100 TeV. We present a novel approach to extract
model-independent constraints on the triple and quartic Higgs self-coupling by
investigating triple Higgs-boson hadroproduction at a center-of-mass energy of
100 TeV, focusing on the $\tau \tau b \bar{b} b \bar{b}$ channel that was
previously overlooked due to a supposedly too large background. It is thrown
into sharp relief that the assist from transverse variables such as $m_{T2}$
and a boosted configuration ensures a high signal sensitivity. We derive the
luminosities that would be required to constrain given deviations from the
Standard Model in the Higgs self-interactions, showing for instance that a
$2\sigma$ sensitivity could be achieved for an integrated luminosity of 30
ab$^{-1}$ when Standard Model properties are assumed. With the prospects of
combining these findings with other triple-Higgs search channels, the Standard
Model Higgs quartic coupling could in principle be reached with a significance
beyond the $3\sigma$ level.
