Compute $\lim_{s\to 0} \left(\int_0^1 (\Gamma (x))^s\space\mathrm{dx}\right)^{1/s}$
Check my answer here to find a proof of the following:
If $\mu$ is a positive measure on a space $X$, $\mu(X) = 1$ and $\|f\|_p$ is finite for some $p$ then: $$ \lim_{p \to 0} \|f\|_{p} = \exp\left(\int_X \log|f| \,d\mu\right) $$
Mathematica suggests that $\|\Gamma\|_{1/2}$ is finite, but I haven't proved this yet. (Edit: See the comment by @DavidMoews below for a proof.)
Check this answer here to find that:
$$ \int_0^1 \log \Gamma(x) \,dx = \dfrac{1}{2}\log(2\pi) $$
And conclude that the limit you're after is $\sqrt{2\pi}$.