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 τ τb ¯ bb ¯ 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 T 2 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 2σ 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 3σ level.