Am I right in thinking this is not possible since both are known to be transcendental?
Also, $e^{i\pi}+1=0$ suggests this is not possible - we can not isolate $e$ or $\pi$ from this since it involves taking a log at some point, thus "cancelling" $e$.
The answer is: This is unknown at the moment. They both are transcendental, but it is unknown if they are algebraically independent. If you want to know more about this, you can read this Mathoverflow thread, or this Wikipedia article on Schanuel's conjecture.