Is there any importance in problems whose witness for membership in a set, cannot be bounded by a polynomial?
If we remove the condition on the size of the witness, then a problem is in the resulting class NANY iff it is recursively enumerable. 

Indeed, if $L$ is in NANY, then given $x$, we can go over all possible $y$, and for each of them run the machine that decides whether $y$ is a witness for $x \in L$.

In the other direction, suppose that the Turing machine $M$ halts on $x$ iff $x \in L$. We can use the transcript of a halting computation as a witness that $x \in L$.