What is known about irrationality of $\pi e$, $\pi^\pi$ and $e^{\pi^2}$?



Irrationality of $ \pi e, \pi^{\pi}$ and $e^{\pi^2}$
I believe most such questions are still very far from being resolved. 


Apparently, it is not even known if $\pi^{\pi^{\pi^\pi}}$ is an integer (let alone irrational).


