Gottfried, if the definitions agree on finite arguments then they also agree in the limit, this is a triviality. Everything else is a matter of taste.
If we define
\( \exp^0(x)=x \)
\( \exp^{n+1}(x)=\exp^n(\exp(x)) \)
then of course also
\( \exp^{n+1}(x)=\exp(\exp^n(x)) \)
and vice versa.
If you want, you can prove that by induction.
If we define
\( \exp^0(x)=x \)
\( \exp^{n+1}(x)=\exp^n(\exp(x)) \)
then of course also
\( \exp^{n+1}(x)=\exp(\exp^n(x)) \)
and vice versa.
If you want, you can prove that by induction.