I know of the continuum hypothesis (CH) and how it was proven to be unprovable under ZFC, but this was after Gödel's incompleteness theorems. And in fact Gödel (and Paul Cohen) were the ones who proved this. So before the incompleteness theorem, was the notion of proving things unprovable not an inkling in any mathematician's head?
The parallel postulate is a good example.
Another one is that the proof of any theorem is also a proof of the provability of the theorem.
For one more, Turing, Kleene, and Gödel all worked on proving that a certain problem couldn’t be solved via an algorithm, but didn't solve it before the Incompleteness Theorems came out. You can read about the problem here, but it basically asked if there was an algorithm that could prove any statement from the axioms of arithmetic. It’s called the Entscheidungsproblem, which is German for “the decision problem.”
This last example is really important because it’s what got mathematicians thinking about unprovability in the 20th century, and was published in 1928. This problem was originally posed as “find an algorithm” because previously to this problem people didn’t seriously consider the idea that the answer might be “you can’t do that.” Gödel’s work on this problem is part of what influenced his thinking on the Incompleteness Theorems, published in 1931. The problem wouldn’t be solved until Turing’s 1936 paper though.
Another problem that was foundational was the Halting Problem. This problem is closely related to both incompleteness and Hilbert’s problem. It was also solved by Alan Turing in 1936.