How do we prove $e^{\frac{1}{e}}$ is irrational ? Also how do we show it is transcendental ?
The number $\eta = \exp(\exp(-1))$ occurs naturally in the context of tetration and power towers.
Let $h(x) = x^{x^{x^{...}}}$ then $x = \eta$ is the largest real $x$ such that $h(x)$ converges. Btw $h(\eta) = e$.
I know $\exp(\pi)$ is irrational (even transcendental) and this looks similar.
Im not even sure the Geldfond-Sneider theorem helps here, or the $h$ function.
To give you an example of when an irrational number raised to an irrational power is rational, here goes:
$$e^{\ln(2)}=2$$
Can you believe me when I say $\ln(2)$ is irrational, hence I have created a counter-example to your logic.
However, this problem is easily misunderstood as well:
$$e^m, m=0,1,2,3,\dots$$
Whether or not that is rational or irrational is not so easily found, for example, $(\sqrt{2})^2=2$ so even if the base is irrational, the result may be rational.
One knows that $\sqrt[b]a=\frac pq$ is irrational if $b$ is a positive whole number and $a$ is not a power of $b$ because you can't write it as a rational fraction $\frac pq$, power both sides by $b$, and get $a$.
Determining such simple things are a lot harder than it seems.