TITLE: Is there an accepted definition of $(\infty,\infty)$ category?
QUESTION [79 upvotes]: For probably twenty years, category theorists have known of some objects in the Platonic universe called "(weak) $\infty$-categories", in which there are $k$-morphisms for all $k\in \mathbb N$, with some weak forms of composition and associativity and ....  It was recognized a bit later that it is worth recording two numbers for each category: an $(m,n)$-category for $m,n \in \mathbb N\cup\lbrace \infty\rbrace$ with $n\leq m$ has morphisms up to dimension $m$ (and above that only equalities), but everything above dimension $n$ is invertible (in some weak sense).  Thus an $(\infty,0)$-category is a "higher groupoid", which Grothendieck's homotopy hypothesis says should be the same as a homotopy type.  (Many people take the homotopy hypothesis to be a definition; others prefer to take it as a check, so that a definition is good if the homotopy hypothesis is a theorem.)  Just to confuse the language, an $(\infty,1)$-category is what many people now call by the name "$\infty$-category".  In any case, for quite a while these objects were expected to exist but definitions were not available (or perhaps too available, but without enough known to decide which was the correct one).
More than a decade ago, Leinster provided A survey of definitions of n-category (Theory and Applications of Categories, Vol. 10, 2002, No. 1, pp 1-70.).  Much more recently, Bergner and Rezk have begun a Comparison of models for (∞,n)-categories (to appear in Geometry and Topology).  These works tend to focus on $(\infty,n)$-categories with $n<\infty$ (although some of Leinster's definitions do include the $n=\infty$ case).
My impression, almost certainly very biased by the people that I happen to hang out with, is that by now it is known that "$(\infty,0)$-category" might as well mean "Kan simplicial set" and "$(\infty,1)$-category" might as well mean "simplicial set in which all inner horns are fillable".  By "might as well mean", I mean that there are many reasonable definitions, but all are known to be Quillen-equivalent.  For $n< \infty$, "$(\infty,n)$-category" can mean "iterated complete Segal space", for example.  It is not clear how to take $n=\infty$ in this approach.
A very tempting definition of "$(\infty,n)$-category", which I believe is discussed in Lurie's (to my knowledge not yet fully rigorous) paper On the Classication of Topological Field Theories, is that an $(\infty,n)$-category should be an $(\infty,1)$-category enriched in $(\infty,n-1)$-categories.  This is based on the observation that the coherence axioms of being an enriched category only require invertible morphisms above dimension $1$.  Thus as soon as you've settled on a notion of "$(\infty,1)$-category" (this seems to have been settled) and a notion of "enriched" (settled?), you get a notion of "$(\infty,n)$-category".  This approach is developed in Bergner–Rezk, for example.
In any case, it is less clear to me how to use these inductive definitions to get to $(\infty,\infty)$.  Let $\mathrm{Cat}_n$ denote the $(\infty,n+1)$-category of $(\infty,n)$-categories.  Then there are clearly inclusion functors $\mathrm{Cat}_n \to \mathrm{Cat}_{n+1}$, but the colimit of this sequence does not deserve to be called $\mathrm{Cat}_\infty$, since any particular object in it has morphisms only up to some dimension.  I have seen it proposed that one should instead take the (inverse) limit of the "truncation" functors $\mathrm{Cat}_{n+1} \to \mathrm{Cat}_n$; I am unsure exactly how to define these, and even less sure why this deserves the name $\mathrm{Cat}_\infty$.
Thus my question:

Is there by now an accepted / consensus notion of $(\infty,\infty)$-category?  For instance, if I have decided on my favorite meaning of $(\infty,1)$-category, is there some agreed-upon procedure that produces a notion of $(\infty,\infty)$-category?  Or are there known comparison results that can push all the way to $n=\infty$?

A closely related MathOverflow question was asked a little less than a year ago (or via Wayback Machine here), but received no answers.
Also, I am certainly unaware of many papers in the literature, and likely I have misrepresented even those papers I mentioned above.  My apologies to all parties for doing so.  It is precisely because of these gaps in my knowledge of the literature that I pose this question.

REPLY [8 votes]: This is more of a general comment on the direction of multiple category theory rather than an answer to the question but I hope it will be useful in directing attention to the origins of some of these questions in homotopy theory and for suggesting lines, or a line,  for further development. In this I will possibly still continue Out of Line! 
I looked at the Berner-Rezk paper referred to in the above, and found the following diagram for compositions in $2$-categories. 
 (source)
Compare this with the simplicity of describing the following in a double category. 
 (source)
One writes this as an array $(a_{ij})$ with composition $[a_{ij}]$, and this notation is easily extended to $n$ dimensions without using operads, though these might be useful in the cubical context.  Note also that an inverse in a specific direction is easy to envisage. 
This is why I have found it very difficult to conjecture and prove results in the globular setting, while the cubical setting has allowed a new approach to basic algebraic topology using filtered spaces and homotopy classes of mappings. 
My writing to Alexander Grothendieck in 1982 did stimulate his writing of "Pursuing Stacks" in 1983, but he noted that the model of $\infty$-groupoids did not model all homotopy types, and proceeded to look for weak categorical models, which are clearly important. But when in 1985 (or 6)  I told him of Loday's work that (strict) $n$-fold groupoids model homotopy $n$-types, he exclaimed: "That is absolutely beautiful!"  This work relates to classical parts of homotopy theory, such as $n$-ads, and the Blakers-Massey and Barratt-Whitehead theorems, and gives some new explicit calculations of  homotopy $n$-types in a nonabelian context. 
Another aspect of Loday's work is his notion of "resolution" of a space: the first stage resolves in one direction, the second in a different direction, and so on, ending up with a resolution of spaces by $n$-cubes of fibrations, which has been clearly explained by Richard Steiner, J. London Math. Soc. (2)  {34} (1986) 169--176. 
What I am arguing for is an eclectic approach, in which the advantages and difficulties of various approaches are considered. 
Update: I'll add a reference to J.C. Morton, "cubical $n$-categories and finite-limits theories", arxiv: 1001.2628, and some further references there.  Another relevant paper is arXiv:1204.5101 [pdf, ps, other]
n-Fold groupoids, n-types and n-track categories, 
David Blanc, Simona Paoli

September 1, 2014    Here is a link to a presentation I gave at the IHP, Paris, June 5, 2014 on Intuitions for cubical methods in nonabelian algebraic topology) (Wayback Machine). 
December 26, 2015 I would also like to add this presentation in June 2015 at Aveiro on 
A philosophy of modelling and computing homotopy types (Wayback Machine), 
which illustrates  the aim of doing specific nonabelian colimit computations in homotopy theory. 
October 26, 2016  A writeup of part of the above presentation is available on the arXiv as Modelling and Computing Homotopy Types:I.