Higher-dimensional Catalan numbers? One could imagine defining various notions of higher-dimensional Catalan numbers,
by generalizing objects they count.
For example, because the Catalan numbers count the triangulations of convex polygons,
one could count the tetrahedralizations of convex polyhedra, or more generally, triangulations of
polytopes.  Perhaps the number of triangulations of the $n$-cube are similar to the Catalan
numbers?
Or, because the Catalan numbers count the number of below-diagonal monotonic paths
in an $n \times n$ grid, one could count the number of monotonic paths below
the "diagonal hyperplane" in an $n \times \cdots \times n$ grid.
I would be interested in references to such generalizations, which surely have been considered—I am just not looking under the right terminology.
I would be particularly grateful to learn of generalizations that share some of the
Catalan number's ubiquity.  Thanks for pointers!
 A: The closest thing I can think of to what you want is triangulations of even dimensional cyclic polytopes. These are nice because, unlike triangulations of the cube, they all use the same number of simplices. See the above linked paper for more.
There are a number of other important generalizations of Catalan numbers, but these are the only ones I would particularly call higher-dimensional.
A: Two papers that both address your question along similar lines are:
(1) Free n-category generated by a cube, oriented matroids, and higher Bruhat orders by Karpranov and Voevodsky and 
(2) Arrangements of hyperplanes, higher braid groups and higher Bruhat orders by Manin and Schechtman
Maps between higher Bruhat orders and higher Stasheff-Tamari posets by H. Thomas serves as a follow-up and fills in in some of the details, but has a different flavor. 
Briefly, paper (1) defines (among other things) a poset $S(n,k), 0 \leq k \leq n$, with $|S(n,2)|$ equal to the $n^{th}$ Catalan number. The objects of this poset are defined in 3 ways (and shown to be equivalent): combinatorially (via cyclic polytopes), as "homotopies of homotopies" and as elements of "the free $(n-k)$-category on the $n$-simplex". 
So if you like these definitions and think they are natural, $|S(n,k)|$ for $k >2$ gives you "higher dimensional" Catalan numbers.
Paper (2) does something nice, too, but I can't remember what. 
A: The Catalan numbers count the standard tableaux of shape (n,n). One way to generalize the catalan numbers is to instead consider the standard tableaux of shape (n,n,n) giving the "three dimensional catalan numbers" and in general you can count the standard tableaux of shape (n,n,n,...,n)=(n^m). 
The higher dimensional catalan numbers have come up recently for me in studying a particular case of the cyclic sieving phenomenon. The standard q-analogue of the higher dimensional catalan numbers form polynomials that, when evaluated at primitive roots of unity, count the fixed point set of the action of the cyclic group generated by the "promotion operator" on rectangular tableaux. 
If you are interested, you can find details of this in a paper by Rhoades.
Rhoades,B. Cyclic sieving, promotion, and representation theory. J. Combin. Theory Ser. A 117, 1 (2010),38-76.
A: I am interested in nested sequences of Dyck paths. I rotate your picture and think of a Dyck path as a sequence of steps $(1,1)$ and $(1,-1)$ which stays above $y=0$. A nested sequence is a sequence of $k$ Dyck paths such that if $1\le i<j\le k$ then the $i$-th path is never above
the $j$-th path. I don't know of any interpretation in terms of polytopes.
A: One relatively direct generalization is to interpret the $n$-th Catalan number as the multiplicity of the trivial representation in the $2n$-th tensor power of the standard $2$-dimensional representation of $SU_2(\mathbf{C})$, and to consider instead the multiplicity of the trivial representation in the $kn$-th tensor power of the standard $k$-th dimensional representation of $SU_k(\mathbf{C})$.
(On the other hand, I don't known which other properties of the Catalan numbers will have any analogue here...)
A: When you count the number of positively directed paths from $(0,0,\dots,0)$ to $(n,n,\dots,n)$ that lie in the region $x_d\le x_1+\cdots+x_{d-1}$, you can project to the plane $(x_d,x_1+\cdots+x_{d-1})$ and find that you need the number of planar paths from $(0,0)$ to $(n,n(d-1))$ which stay above the line $x=y$, and which have $n$ vertical steps of each color {1,...,d-1}. So the answer comes to be $$\left(\binom{nd}{n}-\binom{nd}{n-1}\right)\binom{n(d-1)}{n,\dots,n}=\frac{n(d-2)+1}{n(d-1)+1}\frac{(nd)!}{(n!)^d}.$$
See also this previous question for enumerating lattice paths below a line.
On the other hand, one way to interpret Catalan numbers as lattice paths below the diagonal is to look at it as counting the number of standard Young tableaux of shape $(n,n)$. So a natural generalization is for example the number of standard Young tableaux of shape $(n,n,\dots,n)$. This corresponds to the region $x_1\le x_2\le\cdots\le x_d$, and can be counted using the hook-length formula. See my answer here.
A: This is not really an answer, but I'm just wondering if there are generalizations (in $2$-dimensional setting) of the form: number of paths from $(0,0)$ to $\left(n,\left\lfloor mn \right\rfloor\right)$ that stay below the line $y=mx$? And would that have other interpretations?
A: There's a relation between Associahedra and noncrossing partitions as outlined in this paper:
Rational associahedra and noncrossing partitions
Sloane Encylopedia has along list of facts about Catalan Numbers
Wait.  These are not higher dimensional.
