Limit of The Tower (Strike Back)

$\large e^{-e}$

let $f(x) = a^x$ , if $f^n(x)$ converges to $L$ , then $f(L) = a^L = L, f'(L) = a^L \ln a = \ln L$ .

when $x \approx L$ we approximate $f(x)$ using its tangent at $x=L$ , $\begin{aligned} f(x) &\approx L + f'(L)(x-L) \\ &= L + (\ln L)(x-L) \\ f(L+\varepsilon) &\approx L + (\ln L) \ \varepsilon \\ f^2(L+\varepsilon) &\approx L + (\ln L)^2 \ \varepsilon \\ f^3(L+\varepsilon) &\approx L + (\ln L)^3 \ \varepsilon \\ \end{aligned}$ for $f^n(L+\varepsilon)$ to converge, we must have $\begin{aligned} -1&<\ln L<1 \\ e^{-1}&<L<e \\ (e^{-1})^\frac{1}{e^{-1}}&<a<e^\frac{1}{e} \ (\because a=L^\frac{1}{L} \text{ increases monotonically between } e^{-1}<L<e)\\ e^{-e}&<a<e^\frac{1}{e} \end{aligned}$

p/s: im not sure if this solution is valid, so the result is double checked using computer program, which seems to support my answer.