Which Grows Faster: Factorial or Double Exponentiation
You can find the result by taking a $\log$. Hence:

$$\log(2^{3^n}) = 3^n$$
$$\log(n!) \leqslant \log(n^n) = n\log n$$

(In the latter equation, we have used the fact that $n! \leqslant n^n$, as you note in the question.)

Of course $3^n$ grows faster than $n \log n$. 
As $\log$ is an increasing function, we can say $2^{3^{n}}$ grows faster than $n^n$, and also $n!$.