I don't know whether you‘re going to like it, but the most obvious example of a problem that is undecidable even given an oracle for the halting problem is the

I don't know whether you‘re going to like it, but the most obvious example of a problem that is undecidable even given an oracle for the halting problem is the next halting problem up: i.e. given a Turing machine that has an oracle for the halting problem, does it halt?