I've seen many different notions of $\infty$-categories: actually I've seen the operadic-globular ones of Batanin and Leinster, and the opetopic, and eventually I'll see the simplicial ones too. Although there are so many notions of $\infty$-category, so far I've only seen the following examples:
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$\infty$-groupoids as fundamental groupoids topological spaces;
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$(\infty,1)$-categories, mostly via topological example and application in algebraic geometry (in particular in derived algebraic geometry);
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strict $(\infty,\infty)$-categories, and their $n$-dimensional versions, for instance the various categories of strict-$n$-categories (here I intend $n \in \omega+\{\infty\}$).
There are other examples of $\infty$-categories, especially from algebraic topology or algebraic geometry, but also mathematical physics and computer science and logic?
In particular I am wondering if there's a concrete example, well known, weak $(\infty,\infty)$-category.
(Edit:) after the a discussion with Mr.Porter I think adding some specifications may help:
I'm looking for models/presentations of $\infty$-weak-categories for which is possible to give a combinatorial description, in which is possible to make manipulations and explicit calculations, but also $\infty$-categories arising in practice in various mathematical contexts.
Best Answer
As per Todd's suggestion I am posting this as an answer.
The $(\infty, n)$-category of bordisms is an important example for many reasons, the most imporant of which is its role in the Baez-Dolan cobordism hypothesis. There are several constructions of it, but one of the more modern ones, in the language of $(\infty,n)$-categories ($n$-fold complete Segal spaces), is given in Jacob Lurie's On the Classification of Topological Field Theories.