Are there decision problems outside of NP? Consider any problem in NP-hard, then it has a polynomial reduction from a problem in NP in polynomial time. Though, it isn't clear by this definition whether there are decision problems in NP-hard that are not in NP.

Are there any known ones?
The nondeterministic time hierarchy theorem shows that, e.g., $\mathrm{NP}\subsetneq\mathrm{NEXP}$, so any $\mathrm{NEXP}$-complete problem is $\mathrm{NP}$-hard but not in&nbsp;$\mathrm{NP}$.  For a more extreme example, the halting problem is $\mathrm{NP}$-hard but not in&nbsp;$\mathrm{NP}$.