I know that there are analytic functions whose composition with itself is the exponential function, the so-called functional square root of the exponential function, with the additional property that it is real on the real line. 


Is a similar property possible for a holomorphic function that interpolates the tower function?


Tower function on the positive integers is defined recursively by $f(n+1) = e^{f(n)}, f(1) = 1$. 



What's a natural candidate for an analytic function that interpolates the tower function?
The function you want grows too fast to be interpolated by usual method, but there exists an [iterative solution with Cauchy integrals by Dmitry Kouznetsov and Henryk Trappmann](http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/)


If you relax the condition so to find a solution for $f(x+1)=a^{f(x)}$ such that $$a \le e^{1/e} $$ then there are multiple expressions for your function:


$$f(x)=\sum\_{m=0}^{\infty} \binom xm \sum\_{k=0}^m \binom mk (-1)^{m-k}\exp\_a^{[k]}(1)$$


$$f(x)=\lim\_{n\to\infty}\binom xn\sum\_{k=0}^n\frac{x-n}{x-k}\binom nk(-1)^{n-k}\exp\_a^{[k]}(1)$$


$$f(x)=\lim\_{n\to\infty}\frac{\sum\_{k=0}^{2n} \frac{(-1)^k \exp\_a^{[k]}(1)}{(x-k)k!(2n-k)!}}{\sum\_{k=0}^{2n} \frac{(-1)^k }{(x-k) k!(2n-k)!}}$$


$$f(x)=\lim\_{n\to\infty} \log\_a^{[n]}\left(\left(1-\left(\ln \left(\frac{W(-\ln a)}{-\ln a}\right)\right)^x\right)\frac{W(-\ln a)}{-\ln a}+\ln \left(\frac{W(-\ln a)}{-\ln a}\right)\exp\_a^{[n]}(1)\right)$$


Always here the number in square brackets designates n-th iteration and $W(x)$ is the Lambert's function.


There is also an expression for inverse function:


$$ f^{[-1]}(x)=\lim\_{n\to\infty} \frac{\ln \left(\frac{\frac{W(-\ln a )}{\ln a}+\exp\_a^{[n]}(x)}{\frac{W(-\ln a)}{\ln a}+\exp\_a^{[n]}(1)}\right)}{\ln \ln \left(\frac{W(-\ln a)}{-\ln a}\right)}$$


