What would it mean if there was a link between e and $\pi$? It is not even know if $\pi+e$ is rational, and the same is true for other similar expressions involving $\pi$ and $e$, but does this have an impact?
If it were, for example, proven that $\pi=ae$ or $\pi=a+e$ for a rational number $a$ would there be any consequences beyond it being a bit weird?

If $\pi e$ were rational, the limit points of $\exp(n! i)$ as $n \to \infty$ would be a finite set of roots of unity.
EDIT: Note that from $e = \sum_j 1/j!$ we get $n!e = (integer) + r$ where 
$$0 < r = \sum_{j=n+1}^\infty \dfrac{n!}{j!} < \sum_{k=1}^\infty \dfrac{1}{(n+1)^k} = \dfrac{1}{n}$$
so $\exp(2 i n!\pi e) \to 1 $.  If $\pi e = a/b$ where $a$ and $b$ are positive integers, then $\exp(2in! a) = (\exp(2in! \pi e))^b \to 1$ so any limit point of $\exp(in!)$ is a $2a$'th root of unity.
