  We study di-Higgs and tri-Higgs boson productions at a muon collider as
functions of the modification of the muon Yukawa coupling resulting from new
physics parameterized by the dimension 6 mass operator. We show that the
di-Higgs signal can be used to observe a deviation in the muon Yukawa coupling
at the 10 % level for $\sqrt{s} = 10$ TeV and at the 3.5 % level for $\sqrt{s}
= 30$ TeV. The tri-Higgs signal improves the sensitivity dramatically with
increasing $\sqrt{s}$, reaching 0.8 % at $\sqrt{s} = 30$ TeV. We also study all
processes involving Goldstone bosons originating from the same operator,
discuss possible model dependence resulting from other operators of dimension 6
and higher, and identify multi-Higgs productions and one additional process as
golden channels. We further extend the study to the two Higgs doublet model
type-II and show that di-Higgs and tri-Higgs signals involving heavy Higgs
bosons can be enhanced by a factor of $(\tan \beta)^6$, which results in the
potential sensitivity to a modified muon Yukawa coupling at the $10^{-6}$ level
already at a $\sqrt{s} = 10 $ TeV muon collider. The results can be easily
customized for other extensions of the Higgs sector.
