One thing that might interest you is my result with Clark Barwick which gives an axiomatiation + uniqueness result for the homotopy theory of higher categories: 
http://arxiv.org/abs/1112.0040
(i.e. $(\infty,n)$-categories). 
This axiomatization includes several variants of $(\infty,n)$-category for finite n, such as what you mention in your question.

As Charles Rezk mentions, the inclusion of $(\infty,n)$-categories into $(\infty, n+1)$-categories has both a left and right adjoint and this gives rise to two towers of homotopy theories of higher categories. The limits of these towers give two potential models of $(\infty,\infty)$-categories which are not equivalent.

One consequence of the unicity result is that these towers are essentially uniquely defined and are essentially model independent (provided the models satisfy our axioms). So in that sense there are these two canonical (established?) choices for the homotopy theory of $(\infty,\infty)$-categories. 

I know Clark and I have discussed this idea with many people, but the idea is not new. I think one of the important parts of this story is Eugenia Cheng's theorem from her paper "An omega-category with all duals is an omega groupoid". Her result applies in the tower using the left adjoints, the "coinductive" version. There an $(\infty, \infty)$-category can be tought of as a sequences of $(\infty, n)$-categories, where each truncates to the previous theory. Cheng's result implies that in such a higher category if you have all duals you are an $\infty$-groupoid.

I prefer the other limit, the limit of right adjoints. There an $(\infty,\infty)$-category is a sequences of $(\infty,n)$-categories where the previous is the maximal $(\infty,n-1)$-category. For some reason this seems more natural to me, though I know of others who disagree. 
This version includes one of my favorite examples: the infinite cobordism category. There are cobordisms and cobordisms between cobordisms and cobordisms between these and so on forever. This is an object in the tower of right adjoints which has duals for all objects, but which is not an $\infty$-groupoid. The limit of right adjoints fails Cheng's theorem. It also has a sort of inductive notion of equivalence, rather than coinductive.

Which is the "right" notion of $(\infty,\infty)$-category probably depends on taste and what you want to do with the notion. Both are useful. I would also love to know if there are any theories in between these two!

You can be more explicit about these models too. Dominic Verity has a model of higher categories based on "weak complicial sets". There are $(\infty,n)$-versions of this and also $(\infty, \infty)$-versions. 
One of the conjectures that Clark and I made at the end our paper is that some variant of Dominic's $(\infty,n)$-theory satisfies our axioms (Dominic, Emily Riehl, and I have a partial sketch of this, so hopefully the truth of this conjecture will be known... soon?). If that is true, then it is straighforward to show that the Dominic's $(\infty,\infty)$-version of weak complicial sets is a model of the tower of right adjoints, the one which includes the infinite bordism category. So there are also concrete models of these theories.

There are also others whose work will yield explicit models of this. Rune Haugseng's work was already mentioned. Jeremy Hahn's (upcoming?) work will provide nice model of both limits. I am sure there are many ways to model these two theories.