This title probably seems strange, so let me explain.

Out of the several different ways of modeling $(\infty, n)$-categories, complicial sets and comical sets allow $n = \infty$, providing mathematical definitions of $(\infty, \infty)$-categories. I've asked people a few times for interesting examples of $(\infty, \infty)$-categories that could fit into these definitions, and I've always gotten the answer: the $(\infty, \infty)$-category of (small) $(\infty, \infty)$-categories.

This is not a bad example, and I think it's cool, but I would like to know what kinds of examples are out there other than just categories of categories. For example, for $(\infty, n)$-categories with $n$ finite, "non-categorical" examples include $(\infty, n)$-categories of bordisms as well as the Morita $(\infty, n)$-category of $E_{n-1}$-algebras in an $(\infty, 1)$-category: people care about bordisms and $E_{n-1}$-algebras before learning that they have this higher-categorical structure.

I'm interested in hearing about examples like these for $(\infty, \infty)$-categories. It doesn't matter a lot to me whether something's been rigorously shown to be an example of one of these models or not; and maybe your favorite example is a different kind of $(\infty, \infty)$-category, such as the ones discussed in Theo's question from several years ago; that's also welcome.

What would be really neat is an example of a new phenomenon at the $n = \infty$ level, so an example of an $(\infty, \infty)$-category that's not similar to an $(\infty, n)$-category example for any $n$, but that seems like a lot to ask for.

In addition to Theo's question that I linked above, this question by Alec Rhea and this question by Giorgio Mossa are also relevant, asking similar questions for $n$ finite.

towerin her PhD thesis. $\endgroup$7more comments