A "natural" decidable problem not in $\mathsf{NP}$?
From an answer to a related question on NP-hard problems which are not contained in NP: probably the most natural example is determining whether two regular expressions (including the Kleene star for arbitrary repetition, and a squaring operation to allow compact expressions of very large fixed numbers of repetitions) are equivalent. The resulting problem is EXPSPACE complete. Because EXPSPACE contains NEXP, which contains NP strictly (by the time hierarchy theorem), this is a decideable problem which is not in NP.