We show that the distribution of the spectral maximum of monotonically independent self-adjoint operators coincides with the classical max-convolution of their distributions. In free probability, it was proven that for any probability measures $σ,μ$ on $\mathbb{R}$ there is a unique probability measure $\mathbb{A}_σ(μ)$ satisfying $σ\boxplus μ= σ\triangleright \mathbb{A}_σ(μ)$, where $\boxplus$ and $\triangleright$ are free and monotone additive convolutions, respectively. We recall that the reciprocal Cauchy transform of $\mathbb{A}_σ(μ)$ is the subordination function for free additive convolution. Motivated by this analogy, we introduce subordination functions for free max-convolution and prove their existence and structural properties.