After reading the question and various good answers on the post
I wonder if $3! \cdot 5! \cdot 7! \cdots (2n+1)!$ would evaluate to a factorial of some expression of $n$.
$n=1$, ans $=3!$
$n=2$, ans $=6!$
$n=3$, ans $=10!$ (the one sought in the related post)
What about higher values of $n$?