TITLE: Evaluating $Q = \sqrt{1!+\sqrt{2!+\sqrt{3!+\dots}}}$
QUESTION [27 upvotes]: Consider the nested radical
$$Q = \sqrt{1!+\sqrt{2!+\sqrt{3!+\sqrt{4!+\sqrt{5!+\sqrt{6!+\dots}}}}}}\, .$$
I'm certain the above nested root converges, considering $(x!)^{2^{-x}} \to 1$ (is this a sufficient condition to conclude convergence?) I calculated $Q$ to quite a few digits, and to my surprise, found something that was off by less than $1 \times 10^{-9}$:
$$A=\left(\frac{e^{-\pi}}{\sin(\frac\pi{12})}\right)^{\sqrt{5}}.$$
Which isn't pretty, but it is very close to $Q$, and may be even closer (or, less excitingly, less close) if I calculated $Q$ to more digits. Unfortunately, $Q$ grows insanely slowly, so I can't really do that. Whether this actually converges to this strange number or not, is there any hope in finding a closed form for $Q$?

REPLY [3 votes]: Just for the fun of it !
I gave a friend of mine the constant as printed in sequence $A099876$ at $OEIS$ and he came back with the approximation
$$\frac{2-\sqrt{2}+2 \sqrt{3}-3 e-2 \pi -5 \pi ^2+\log (2)+2 \log (3)}{2 \sqrt{2}+8
   \sqrt{3}-9 e+8 \pi -6 \pi ^2+6 \log (2)+6 \log (3)}$$ which is in a relative error of $8.14 \times 10^{-18}$%.
Not very nice !