Set of all decision problems solvable by a Turing machine, i.e. that decide if a string belongs to a recursive language.
Or in other words: there is no Turing machine that always halts for every input with the yes/no output.
Every undecidable problem must obviously have an infinite number of "possibilities of stuff you can try": if there is only a finite number, then you can brute-force it.
Lists of undecidable problems.
Coolest ones besides the obvious boring halting problem: