on the growth of standard error

The absolute
value of e grows as L grows, but by how much?  It seems statistical
theory claims it grow by an order of the square root of L.

Assuming you want the standard deviation for the number of successes,
given p=0.5:

#exact
0.5 * sqrt (n)

#numerical approximation
sd (rbinom (1e6, n, 0.5) )

Note that variance should be linear in n.