I'm kind of in the opposite camp. If Schanuel's conjecture is true, then e^iπ = 0 would be the only non-trivial relation between e, π, and i over the complex numbers. And the fact that we already found it seems unlikely.
e^(i theta) = cos theta + i sin theta
That formula gives infinitely many trivial relationships like this due to the symmetry of the unit circle
e^(i 2 pi) = 1
e^(3i/2pi)/i=1
e^(5i/2pi)/i=-1
e^(i 2n pi) = 1 for all n in Z ...
etc