What is (319!)!? This question came to my mind when someone wrote 319!! on my dorm (room 319). I was able to find 319! and it's about 10^661. I used mathematica to compute (n!)! and I was able to go up to 12 and came out around 10^(3,000,000,000). I would like to know how big (319!)! is, I just don't know where to go from here.
 A: Stirling’s formula says:
$$n!\sim \sqrt{2\pi n}\left(\frac ne\right)^n$$
or $$\log_{10} n!\approx n\log_{10}n-\frac{n}{\ln 10}+\frac{\log_{10}(2\pi n)}{2}$$
When $n$ is huge, the last term is irrelevant.
So $$
\begin{align} 
\log_{10}\left(10^{661}!\right)&\approx661\cdot 10^{661}-\frac{1 }{\ln 10}\cdot 10^{661}\\&\approx 6.606\cdot 10^{663}
\end{align} 
$$
More generally, $$\log_{10}(10^m!)\approx (m-\log_{10}e)\cdot 10^m$$
A: If you use the first values you had and plot them, you will notice that
$$\log\Big[\log\big[(n!)!\big]\Big]\sim a n - b$$
A quick and dirty regression gives with $R^2=0.999777$
$$\begin{array}{clclclclc}
 \text{} & \text{Estimate} & \text{Standard Error} & \text{Confidence Interval} \\
 a & 2.24620 & 0.02429 & \{2.18676,2.30565\} \\
 b & 4.99077 & 0.15869 & \{4.60247,5.37906\} \\
\end{array}$$ that we can approximate as
$$\log\Big[\log\big[(n!)!\big]\Big]\sim  n \log(10) - 5$$
$$\log\big[(n!)!\big]\sim  10^n e^{-5}$$
So $$(319!)! \sim \exp\big[7 \times 10^{216}\big]$$
