Natural problem arising from diagonalization?
In some sense the halting problem satisfies this request (I was almost ready to say "There are no such examples!" when I thought of this one.) By doing diagonalization against computable sets, you can of course get sets of very high complexity, but the seemingly-most-natural way of diagonalizing against computable sets gives you a problem that is equivalent to the halting problem. As we now know, the halting problem is equivalent to many other computational problems, such as deciding many properties of TMs, telling whether a Diophantine equation has solutions, the word problem in finitely presented groups, telling whether a high-dimensional knot is the unknot, etc.

I am still ready to say that I would be surprised to see other examples (that aren't just, say, relativized versions of this).