Asymptotic behavior of $\Gamma^{-1}(x)$ For real $x,$ it's well-known that
$$\Gamma^{-1}(x)\sim\frac{\log x}{\log\log x}$$
So a natural question is to bound
$$G(x)=\Gamma^{-1}(x)\frac{\log\log x}{\log x}$$
which of course is 1 + o(1).  Interestingly, its value is near 2 (that is, far from its asymptotic value) for many useful values of x: for example, $1.8<G(x)<2.1$ for $14<x<10^{77}.$
It seems that $G$ has a maximum near 3637.133905003816072848664328 of 2.01119450670696919822787997170113557148977275... and to decrease very slowly thereafter.
Question 1: Is the above the unique maximum?
Question 2: Is there an $x>5$ such that $G(x)<1$?
Question 3: Is there a useful factor or secondary term that makes this approximation more precise for useful values of x?  I'm being intentionally vague on this point—if I knew exactly what I was looking for I probably wouldn't need to ask. :)  For example, had I asked an analogous question about the prime-counting function, telling me about Li would be better than just giving the next asymptotic term $cx/\log^k x.$
 A: I got a better approximation by fiddling with Stirling's formula out of Abramowitz and Stegun, just using the first term $$ \log \Gamma(z) \sim z \log z,  $$ 
take $t = \Gamma(z)$ and the approximation $$ z_1 = \frac{L_1}{L_2} + \frac{L_1 L_3}{L_2^2}, $$
where $L_1 = \log t, \; \; L_2 = \log \log t, \; \; L_3 = \log \log \log t.$
I get $$ z_1 \log z_1 = \log t - \frac{L_1 L_3^2}{L_2^2} + \frac{L_1 L_3}{L_2^2} + smaller  $$ which is an improvement on your folklore result, as the unwanted terms are actually smaller than the next term $z$ in the fuller
$$  \log \Gamma(z) \sim z \log z - z - \frac{1}{2} \log z + \frac{1}{2} \log {2 \pi} + \log \left(1 + \frac{1}{12 z}  + \frac{1}{288 z^2}- \cdots    \right)  $$
You ought to be able to do something with this for your original question.
In particular, it says your $$ G(t) \sim 1 + \frac{\log \log \log t}{\log \log t}, $$ meaning change is so slow that the limit is invisible to a computer, your experimental bounds may well be correct but certainty will be hard to come by. 
