Complexity lower bound of finding the factorial of a number
One approach on this question, related to complexity-theoretic questions, is due to Shub and Smale [1] who proved that if $n!$ is ultimately hard to compute, then $\mathsf{VP}\neq\mathsf{VNP}$ over some field. Their model of computation is the straight-line programs: The goal is to compute $n!$ from the constant $1$, using only additions, subtractions and multiplications. That $n!$ is ultimately hard means that the number of operations needed is asymptotically not bounded by some polynomial in $\log n$. 

For more recent results in this direction (and pointers to the relevant literature), you can read a paper of Cheng [2]. The paper contains also a very brief remark about the standard complexity of the problem, that has to be exponential in $\log n$ (because of the output size), and of the modular variant of it (computing $n! \bmod m$) and its relations with integer factoring.

[1] Mike Shub and Stephen Smale. On the intractability of Hilbert’s nullstellensatz and an algebraic version of “P=NP?”. Duke Math. J., 81:47–54, 1995.

[2]  Qi Cheng, On the Ultimate Complexity of Factorials, Theoretical Computer Science, Volume 326, Issues 1-3, Pages 419-429.