Iterated exponentiation (tetration) to non-integral indices:
Six to eight years ago on the Prodigy network I asked, if 2#3 is 2^(2^2) = 16, then what is 2#(1/3). A mathematician named John George said he didn't know but somebody somewhere had probably done some work on that. A fellow named Arnold Benson found a formula that I used as the basis of some Mathematica computer programs.
Somehow 2#(1/3) got defined as the number x such that x#3 = 2. The definition is just a wild guess. I think the value should be derived from the very definition of tetration (iterated exponentiation), but I can't do it. The original question is still open I guess.