While reading this post, I stumbled across these definitions (Wiki_german)
$e = \lim_{n \to \infty} \sqrt[n]{n\#}$
and
$e = \lim_{n \to \infty} (\sqrt[n]{n})^{\pi(n)}.$
The last one seems interesting, since $ \lim_{n \to \infty} (\sqrt[n]{n})=1$, proven here.
How to prove these?