Lately I've been studying some functions of the form $f_1(f_2(\cdots(f_n(x))\cdots))$ where each $f_i(x) = a_i^x$ for some positive, nonunity, real numbers $a_i$. So for example, if $a_1 = 2$ and $a_2 = 0.3$ the function would be $2^{0.3^x}$. A more elaborate example would be the function, $0.9^{e^{1.01^{0.5^x}}}$. I'm not studying any one particular function, but rather a whole family of them together. Something I've noticed is that they all seem to have at most only one inflection point (alternatively their derivatives have at most one relative extremum). But I cannot seem to prove this result, so it remains an observation.
The problem for me has been that the derivatives, even in the small $n=3$ cases are a little too unwieldy. For anyone who is curious, if we denote $F_m^n(x) = f_m(f_{m+1}(\cdots (f_n(x))\cdots))$ (and trivially let $F_{i+1}^i(x) = x$) then $$\big(F_m^n\big)^{\prime}(x) = \prod\limits_{i=m}^n F_i^n(x)\ln(a_i)$$ and $$\big(F_m^n\big)^{\prime\prime}(x) = \big(F_m^n\big)^{\prime}(x)\times\bigg(\sum\limits_{i=m}^n\ln(a_i)\Big(F_{i+1}^n\Big)^{\prime}(x)\bigg)$$
Setting the latter factor (the one with the sum) in the second derivative equal to $0$ hasn't given me much success. My suspicion is that there is some argument to be made where you could group the factors of $\big(F_m^n\big)^{\prime}(x)$ by whether they are increasing or decreasing functions (it isn't hard to see that a composition of some number of exponential functions will be either always increasing or always decreasing), like so:
$$\big(F_m^n\big)^{\prime}(x) = \Big(\prod\limits_{i=m}^n\ln(a_i)\Big)\times\Big(\prod\limits_{\text{increasing ones}}F_i^n(x)\Big)\times\Big(\prod\limits_{\text{decreasing ones}}F_i^n(x)\Big)$$ and then argue that as $x$ increases from $-\infty$ one of the two latter factors will dominate the other (causing, for example, the whole derivative to increase), and then there will be a point where it switches and the other factor becomes more dominant (causing then an overall decreasing trend).
But of course, I could be completely off base on this. I was wondering if anyone knew of some resources to check out for these kinds of functions, maybe knew a counterexample, or perhaps even knew of an actual proof that each of these kinds of functions have at most one inflection point.