What is known about 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). 


Brownawell and Waldschmidt do have results in these directions which do not rely on Schanuel's Conjecture. The references are M. Waldschmidt, "Solution du Huitième Problème de Schneider," J. Number Theory 5 (1973), 191202. W. D. Brownawell, "The algebraic independence of certain numbers related by the exponential function," J. Number Theory 6 (1974), 2331. The two papers independently prove results along the following lines. (The following version is taken from Brownawell.) Let $\alpha$, $\beta$, and $\gamma$ be nonzero complex numbers with $\alpha$ and $\beta$ both irrational. If $e^\gamma$ and $e^{\alpha\gamma}$ are both algebraic numbers, then at least two of the numbers $$\alpha, \beta, \gamma, e^{\beta\gamma}, e^{\alpha\beta\gamma}$$ are algebraically independent over $\mathbb{Q}$. This theorem has several interesting consequences:
So as a partial answer to this question, at least one of $e\pi$ and $e^{\pi^2}$ is transcendental. 

