Is there a reason we consider $\infty$-categories to be the $\omega^{th}$ step in the 2-internalization inside Cat (or enrichment over Cat if you prefer) process made invertible above some finite ordinal, and don't continue on to higher steps in the recursion? Is there nothing to be gained, or is the $\omega^{th}$ step already mysterious enough that going further is foolhardy?

For example, it seems (very naively) that something like a $(\omega_1,\omega)$-category or higher categories defined up to large cardinals that become invertible at smaller large cardinals might be interesting, or in a $\neg CH$ universe we could ask about $(\omega_1,\mathfrak{c})$-categories and the like. My apologies if this question is trivial, but I couldn't find a discussion/explanation in the literature.