I've seen many different notion of $\infty$-categories, actual I've seen the operadic-globular ones of Batanin and Leinster and the opetopic too and eventually I'll see the simplicial ones too. Although there are so many notion of $\infty$-category so far I've only seen the following examples:

* $\infty$-grupoids 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 example of $\infty$-categories, especially from algebraic topology or algebraic geometry?
In particular I wondering if there's a concrete example, well known, weak $(\infty,\infty)$-category.