How can one demonstrate there is no sequence $X_i$ of sets such that $X_{i+1}' = X_i$ (this is really equality as sets though Turing equivalence would be interesting too).

I know it fails if I relax equality to simply $X_{i+1}' <_T X_i$ as in this question but I'd think there has to be some kind of simple fixed point style argument in this case. Maybe I'm just sleep deprived but I'm blanking on how to show it and even trying the Martin determinacy stuff didn't immediately give me an answer.

Specifically, the reason I care is I need to show that for any $X$ there is a maximum value of $k \in \omega$ such that $(\exists Y)(X = Y^k)$.