In this work, we consider maxima of systems of randomly time-changed Lévy particles. We give a general construction to obtain infinite-dimensional classes $\{Z^α\}$ of stationary max-infinitely divisible (max-id) processes. These classes are indexed by admissible mass functions $α$, which induce state-dependent time changes of the underlying Lévy particles. This gives a generalization of the well-known (Lévy--)Brown--Resnick process $Z^{1}$. In contrast to $α\equiv 1$, the variability of non-constant mass functions $α$ changes the dependence structure of the max-id process and goes beyond the max-stable setting while preserving stationarity. We then explore the extent of the so-called max-domain of attraction (MDA) of a given (Lévy--)Brown--Resnick process $Z^1$, by studying convergence of rescaled maxima of independent copies of $Z^α$ to $Z^1$. Thus, our work combines potential theory for Markov processes and extreme value theory to yield a novel, infinite-dimensional, and interpretable class $\{Z^α\}$ of stationary processes in the MDA of a given (Lévy--)Brown--Resnick process $Z^{1}$. So far, results on the extent of such domains have been scarce in the literature.