Decision problems whose verifier is NP
Suppose you had a problem such that for any $x \in L$, there was a verifier $v$ such that $v$ could be checked against $x$ by a nondeterministic polynomial time algorithm. For a valid $(x, v)$ pair, there is some verifier $v'$ such that it takes polynomial time to check $((x, v), v')$ is a correct verification.

But then, you could simply combine $(v, v')$ into a single witness, and run the check in deterministic polynomial time. 

Thus, the class ${\bf N(NP)}$ you describe is really just equal to ${\bf NP}$.