I was thinking about examples where a problem is NP-hard but was not NP-complete and ILP came to mind.
It is obviously NP-hard but is it NP-complete? I.e., is it in NP? Given a certificate (the alleged minimum), we certainly cannot check if it is the minimum in polynomial time. Therefore, I don't see how we can convert this into a decision problem.
So is ILP NP-hard but not NP-complete?
Also feel free to include examples (apart from the halting problem) of problems which are NP-hard but not NP-complete.