**A**
19 = sqrt(((3!)!)/2 + 1).

I don't know that this amusing formula has any deeper significance. I do have a heuristic explanation that suggests that there are more such formulas than one might guess at first.

The number ((3!)!)/2=360 clearly has many factors, and so can be written
as A·B in many ways. Often A,B are of the same parity,
so can be written as x+y and x-y for some integers x,y,
whence 360=x^{2}-y^{2}. Since 2y=A-B,
we know that y will be small if A and B are close to each other.
In particular, 360=18·20 yields 360=19^{2}-1^{2},
which is equivalent to our formula above.

This kind of thing happens a few more times;
for instance, 71=sqrt(7!+1),
and we can use three 9's to write 603
as the square root of 9!+9^{sqrt(9)}.

The formula 19^{2}-1=(3!)!/2 can be generalized
in a different direction by writing (3!)!/2 as 6!/2!
and noting that for all n>3 we have the identity

As it happens, the identity 71=sqrt(7!+1) above is a consequence of this formula together with the sporadic factorial identity 6!7!=10!.