about transcendental and algebraic numbers.

Question

a few days ago I asked about comparing numbers $e^{\pi^e}$ and $\pi^{e^{\pi}}$, etc. You can see in

How to prove that $\pi^{e^{\pi^e}}<e^{\pi^{e^{\pi}}}$

Now, does anyone know of any result about the transcendence or not of the said numbers?