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
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 $cx/\log^k x.$