Powers of transcendental numbers that lead to integers

Question

For a given real number $x>0$ define $a_k(x)$ as follows for $k \geq 1$: $a_1(x)=x$, $a_{k+1}(x)=x^{a_k(x)}$, so that $a_2(x)=x^x$, $a_3(x)=x^{x^x}$....

Question 1: Is there a explicit transcendental number $x$ such that $a_k(x)$ is an integer for a $k>0$?

Question 2: Let $Y:= \{x \in \mathbb{R} \mid x>0$, $x$ is transcendental and there exists $k >0$ such that $a_k(x)$ is an integer $\}$. Is $Y$ measurable and if yes, what is its measure?

The question is motivated by https://www.youtube.com/watch?v=BdHFLfv-ThQ where it is discussed whether $a_4(\pi)$ is an integer.

Answer 1

For question 2, the answer is that the set is countable, so it is measurable and has measure $0$. Given any $n\gt 1$ and $k\gt 1$ we can see $a_k(x)$ is monotonically increasing with $x$, so we can implicitly invert it. We can find $x$ numerically to whatever precision we want, but generally you won't find a formula for it.

Answer 2

A very simple answer to question $1$

Question $1$: Is there a explicit transcendental number x such that $a_k(x)$ is an integer for a $k>0$?.

Let $k=1$, thus we have $a_1(x) = x$. Thus, you may have $x = \pi, e,$ or $ln(N)$ for any positive natural number $N$. There may be more, but those are just some examples because any transcendental number may be substituted in for $x$.