Think of the factorial as $f(n) = n \odot (n-1) \odot \cdots \odot 2 \odot 1$, where $\odot$ is the binary operator for multiplication, $\cdot$. This suggests exploring replacing $\odot$ with other operators.

$\odot = +$ just results in $(n+1)n/2$. E.g., for $n=10$, the plus-factorial is $55$.

$\odot = -$ results in $-n(n-1)/2$. E.g., for $n=10$, the minus-factorial is $-35$. Here (and below) I am associating to the left, i.e., $((4-3)-2)-1=-2$.

So I explored the binary operation $ a \odot b = \sqrt{a b}$, a geometric-mean operation.
Then the geometric-mean-factorial $gm!(n)$ looks like this:
$$gm!(2) = \sqrt{2 \cdot 1} \approx 1.41421$$
$$gm!(3) = \sqrt{(\sqrt{3 \cdot 2}) \cdot 1} = 6^{\frac{1}{4}} \approx 1.56508$$
$$gm!(4) = \sqrt{(\sqrt{(\sqrt{4 \cdot 3}) \cdot 2}) \cdot 1}=\sqrt{2} \cdot 3^{\frac{1}{8}} \approx 1.62239$$
$$\ldots$$
$$gm!(20) \approx 1.66169$$
The geometric-mean-factorial seems to be approaching a limit that
is unfamiliar to me.
Does anyone recognize this constant from another context?