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Abstract Multi-TeV muon colliders promise an unprecedented potential for exploring the particle-physics energy frontier, and, at the same time, can probe with fantastic accuracy the precise structure of the Standard Model, and in particular of the Higgs boson sector. Here we consider the possibility to measure the loop-suppressed single-, double-, triple-Higgs boson production mediated by the collinear-photon scattering in the channels $$\mu ^+\mu ^-\rightarrow \mu ^+\mu ^-\gamma ^*\gamma ^*\rightarrow \mu ^+\mu ^- \,[H,HH,HHH]$$ μ + μ - → μ + μ - γ ∗ γ ∗ → μ + μ - [ H , H H , H H H ] . We study total rates and kinematical distributions in the Standard Model, and compare them with the corresponding vector-boson-fusion processes $$V^*V^*\rightarrow H,HH,HHH$$ V ∗ V ∗ → H , H H , H H H at muon collisions center-of-mass energies 1.5 TeV TeV. Possible strategies for enhancing the $$\gamma \gamma \rightarrow H,HH,HHH$$ γ γ → H , H H , H H H signal over the dominant vector-boson-fusion production are proposed. The sensitivity of total rates to possible anomalies in the Higgs-boson self-couplings is also discussed.