David Cantrell gives a good approximation of $\Gamma^{-1}(n)$ on this page.
I'll copy the result here in case that page ever goes down:
k = the positive zero of the digamma function, approximately 1.461632
c = $\sqrt{2\pi}/e - \Gamma(k)$, approximately 0.036534
L(x) = $\ln((x+c)/\sqrt{2\pi})$
W(x) = Lambert W function
ApproxInvGamma(x) = L(x) / W(L(x) / e) + 1/2
