Schanuel's conjecture on the transcendence degree of certain field extensions of the rational numbers. In particular: Are \pi and e…

Schanuel's conjecture on the transcendence degree of certain field extensions of the rational numbers. In particular: Are \pi and e algebraically independent? Which nontrivial combinations of transcendental numbers (such as e + \pi, e\pi, \pi^e, \pi^{\pi}, e^e) are themselves transcendental?