Di-Higgs and Tri-Higgs boson signals of muon $g-2$ at a muon collider

We show that new physics explanations of the muon $g-2$ anomaly by the contributions of new leptons mediated by the standard model Higgs boson necessarily lead to large rates for $\mu^+ \mu^- \to hh$ and $\mu^+ \mu^- \to hhh$ irrespectively of details of the model or the scale of new physics. For new leptons with the same quantum numbers as the standard model leptons, cross sections are expected to be about 240 ab for $\mu^+ \mu^- \to hh$ independently of the center of mass energy, $\sqrt{s}$, and about 2.7 ab for $\mu^+ \mu^- \to hhh$ for $\sqrt{s} = 1$ TeV and growing quadratically with $\sqrt{s}$. Predictions for models featuring new leptons with different quantum numbers and for a type-II two Higgs doublet model, where additional Higgs bosons can contribute to muon $g-2$, are also presented.