https://en.wikipedia.org/wiki/Tarski%27s_exponential_function_problem shows a very interesting problem, as for me begging for an undecidability proof (as the Tarski-Seidenberg theorem itself is already very strong, and this would strengthen it a lot more)
But is it decidable with an oracle for arithmetical truth? (Given an arithmetical sentence, this oracle will return whether it is true or false) It sounds like the next best thing -- to be "arithmetically decidable"..
It seems like the problem of whether a constant (say, involving only natural numbers, exponents and basic operations, like e^2 - e^e + 1) is zero or not is at most 0', since we can compute the constant with arbitrary precision and ask whether this program will eventually give an interval not containing zero.