Is there a reason we consider [$\infty$-categories][1] 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.

  [1]: https://ncatlab.org/nlab/show/%28infinity%2Cn%29-category