Since vectors can be thought of as a 1 x n array of elements, and a matrix is an n x m array of elements, then why not continue this sequence to mathematical objects with n x m x p elements and n x m x p ...
What would be the multiplication rules for such an algebra?
I can't think of an application for this (unless it is the same as a tensor?) in which case it is used extensively in electromagnetic fields, relativity, quantum mechanics and particle physics.
Could it be, for instance, by analogy with a linear equation, where a matrix transforms on vector into another:

= 


that a hypermatrix might transform one matrix into another (multilinear algebra?):

= 

Of course we can transform a 3x3 matrix into another 3x3 matrix by multiplying it by a third 3x3 matrix. Would it be more general to multiply it by a hypermatrix?
Or is it possible to have a matrix equivalent of multivectors (Clifford algebra) with two types of multiplication. An inner product which transforms:
real number < vector < matrix < hypermatrix
and an outer product which transforms:
real number > vector > matrix > hypermatrix
What is the relationship between a hypermatrix and a tensor?
Is a hypermatrix just a representation of a tensor?
I think that perhaps a tensor must be square, cuboid, ... In other words it must be the same size in each dimention?
So perhaps not all hypermatrices are tensors?
Rank  representation  element notation 

0  scalar  a 
1  vector  a_{i} 
2  matrix  a_{ij} 
3  hypermatrix of rank 3  a_{ijk} 