I really enjoyed this video on the possibility that $\pi^{\pi^{\pi^\pi}}$ is an integer, but i thought that it was a case of a more interesting general problem.
Given a real $x$ and an integer $y$, what is the minimum non-zero integer $y$ such that the tower $x↑↑y$ is an integer; where $↑↑$ represents tetration (i.e. repeated exponentiation)?
Its clear that for integers $y=1$; and that for the $n$th root of a multiple of $n, y$ is at most $n+1$. For what other real numbers can we obtain $y$? Should we expect $y$ to exist for all real numbers?