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:

$\infty$-groupoids as fundamental groupoids topological spaces;

$(\infty,1)$-categories, mostly via topological example and application in algebraic geometry (in particular in derived algebraic geometry);

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.