First, recognize than the functions you are working with must have some decay for any "efficient" method to be possible. If your functions have rich structure in all modes of the full tensor product, then you can't do better. But many functions of interest have decay in their mixed derivatives, in which case efficient methods are available. The term "separation rank" is sometimes used to refer to how many mixed derivatives are needed.
Sparse tensor grids are a common and rigorously analyzed approach. For appropriate classes of functions, sparse grids reduce storage requirements from $\mathcal O(n^d)$ for a full tensor-product grid to $\mathcal O(C^d n \log^{d-1} n)$ with (usually) small value of $C$. This is a very active field, especially as it pertains to stochastic partial differential equations (SPDEs). These review papers are good sources.
For some problems in dimensions around 6, some physics applications use particle, particle-in-cell, or even naive discretizations of the space (e.g. spherical harmonics to discretize a distribution in angle space at every grid cell in physical space). These methods don't have the quasi-optimality with respect to dimension that the sparse grid techniques have, but they don't require any particular decay (low separation rank).
If you don't need to perform general manipulations of the functions, but instead want to determine some specific properties, there may be other ways to formulate the question to avoid needing to represent it in these ways.
