There’s a real vs theoretical distinction. Turing machines are defined as having infinite memory. Running out of memory is a big issue that prevents computers from solving problems that Turing machines should be able to solve.

The halting problem, a bunch of problems involving prime numbers, a bunch of other weird math problems are all things that can’t be solved with Turing machines. They can all sort of be solved in some circumstances (eg A TM can correctly classify many programs as either halting or not halting but there are a bunch of edge cases it can’t figure out, even with infinite memory).

From what I remember, most researchers believe that human brains are Turing Complete. I’m not aware of any class of problem that humans can solve that we don’t think are solvable by sufficiently large computers.

You’re right that Quantum Computers are Turing Complete. They’re just the closest practical thing I could think of to something beyond it. They often let you knock down the Big Oh relative to regular computers. That was my point though. We can describe something that goes beyond TC (like “it can solve the halting lemma”) but there don’t seem to be any examples of them.