The Fox H-function, as far as I know, is the most general families of functions - encompassing an even larger family of functions than the already very general Meijer G-function. While I've known about these functions for a while, I've never seen a good use of them - are they just generalizations for generalizations sake? Is there ever a time when the use of these functions has given insight into a problem that other assumptions (smooth, continuous, etc.) could not? What fields are they most used in?
Fox–Wright function or even the Fox H-function are very useful on dealing the infinte series expressions whose its coefficient involves for example the gamma functions of linear expressions in the index $n$.
We can't say that the Fox H-function is the most general families of functions, since in fact there are no functions really in the most general families.
This link http://www.dtic.mil/dtic/tr/fulltext/u2/a252517.pdf is a PhD-thesis using H-function in the study of statistical distributions. There are other papers with applications in this area, such as The Distribution of Products, Quotients and Powers of Independent H-Function Variates by Bradley D. Carter and Melvin D. Springer SIAM J. Appl. Math., 33(4), 542–558. (17 pages)
Search in google scholar will give you more papers in this area.