  ${\rm W}^\pm {\rm W}^\pm {\rm H}$ production at hadron colliders through
vector boson scattering is a so far unconsidered process, which leads to a
clean signature of two same-sign charged leptons and two widely separated jets.
This process is sensitive to the ${\rm HHH}$ and ${\rm WWHH}$ couplings and any
deviation of these couplings from their SM predictions serves as direct
evidence of new physics beyond the SM. In this paper we perform a Monte Carlo
study of this process for the $\sqrt{s}=14$ TeV LHC and a $100$ TeV
pp-collider, and provide projections of the constraints on the triple-Higgs and
${\rm WWHH}$ quartic couplings for these environments. In particular, we
consider the impact of pileup on the expected sensitivity in this channel. Our
analysis demonstrates that although the sensitivity to the ${\rm HHH}$ coupling
is rather low, the ${\rm WWHH}$ coupling can be constrained in this channel
within $\sim 100\%$ and $\sim 20\%$ at 95\% confidence level around the SM
prediction at the HL-LHC and a 100 TeV pp-collider, respectively.
