Open-problem statements, with their sources attached.

$e^{\pi^2}$ is transcendental.

$e^e$ is transcendental.

$\pi^e$ is transcendental.

$\pi^{\sqrt{2}}$ is transcendental.

$\pi^{\pi}$ is transcendental.

$\pi^{\pi^{\pi}}$ is transcendental.

$\pi^{\pi^{\pi^\pi}}$ is transcendental.

$\pi^{\pi^{\pi^\pi}}$ is not an integer.

This would follow from $\pi^{\pi^{\pi^\pi}}$ being transcendental, but this formulation is of interest in its own right, as it could in principle be proven by direct computation. Reference:* YouTube

$\log(\pi)$ is transcendental.

$\log(\log(2))$ is transcendental.

$\sin(e)$ is transcendental.

The Catalan constant $G$ is transcendental.