We know that the exponential map is infinitely differentiable; is there a tetration map $TET$ defined on $(0,\infty)$ such that
1) $TET(x) > TET(y)$ when $x > y$
2) $TET(e^x) = TET(x) +1$
3) $TET(x) $ is infinitely differentiable on $(0,\infty)$
4) $TET(1) = 0$
5) (optional) $TET$ has a local taylor series at each point
Note here that $TET(e) = 1, TET(e^e) = 2, TET(e^{e^e}) = 3...$
A real analytic inverse to tetration was constructed in
H. Kneser. Reelle analytische Lösungen der Gleichung $\phi(\phi(x)) =\mathrm{e}^x$ und verwandter Funktionalgleichungen. J. Reine Angew. Math. 187 (1949), 56–67.
Holomorphic analytic continuation of inverse tetration is discussed in
H. Trappmann and D. Kouznetsov. Uniqueness of holomorphic Abel functions at a complex fixed point pair. Aequationes mathematicae 81 (2011), 65–76.