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

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