ct.category theory – Categories $ infty $ higher

Is there a reason we consider $ infty $-categories of being the $ omega ^ {th} $ step in internalisation 2 inside Cat (or enrichment on Cat if you prefer) process rendered invertible over a finite ordinal, and do not proceed to the higher stages of recursion? Is there nothing to gain, or is there $ omega ^ {th} $ already mysterious enough to go further is reckless?

For example, it seems (very naively) that something like a $ ( omega_1, omega) $– a category or higher categories defined up to the great cardinals that become invertible in the greatest cardinals could be interesting, or in a $ neg CH $ universe we could ask questions about $ ( omega_1, mathfrak {c}) $-categories and the like. I apologize if this question is trivial, but I have not found any discussion / explanation in the literature.