Language in $R \setminus \mathit{NP}$
We always measure the running time of a machine with respect to input size. For this reason, your language is in P. Given a sequence of moves, you can easily check in polynomial time that the sequence is valid.
You can construct a language in $R \setminus \mathit{NP}$ using diagonalization. This is the content of the nondeterministic time hierarchy theorem.