TITLE: Concrete example of $\infty$-categories
QUESTION [26 upvotes]: 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.

REPLY [6 votes]: A concrete example of a weak $\infty$-category, but not well studied abstractly, is the  cubical singular complex of a space, preferably with the connections introduced in our  1981 JPAA paper with Philip Higgins. The clear advantage of the cubical setup is the command of multiple compositions, using an easy matrix notation. Thus one can express for the diagram
 (source)
that the big square is the composition of the little squares, by simply writing something like $\alpha=[\alpha_{ij}]$. (How does one do that simplicially, or globularly?) This and higher dimensional versions are useful in expressing algebraic inverses to subdivision, a useful tool in local-to-global problems. Because higher groupoids are nonabelian, unlike higher groups, one can also obtain nonabelian results in algebraic topology. 
The book Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids (EMS, (2011), pdf available from  here) has a large number of uses of algebraic and geometric (e.g. homotopical) conclusions derived from rewriting  such multiple arrays. The main results were conjectured and eventually proved by cubical methods.