What's after EXPSPACE?
EXPSPACE is not the most inclusive computational complexity class.  There's a huge number of complexity classes; see, e.g., the Complexity Zoo for some that have been studied (and in principle there are an unlimited number one could define).

The space hierarchy theorem shows that for each $f(n)$, there are languages that can be decided in space $f(n)$ but cannot be decided in space $o(f(n))$.  If we take $f(n)=2^{2^n}$ (say), then we find there are problems that can be decided in space $2^{2^n}$, but not in space $2^n$, and indeed, are not in EXPSPACE.  So, we can "make up" a complexity class, call it 2EXPSPACE, of problems that can be solved in doubly-exponential space; it follows that 2EXPSPACE is bigger than EXPSPACE.  And so on.  There is an infinite hierarchy of complexity classes, each bigger than the previous.

This also shows that EXPSPACE is not equal to RE (the recursively enumerable problems).