Where the sample correlation is most biased

Idea. The bias is proportional to $\rho (1-{\rho }^2)$: it vanishes at $\rho =0$ (by symmetry) and at $\rho =1$ (a perfect correlation is estimated perfectly), so it peaks in between.

Step 1. $g(\rho )=\rho -{\rho }^3$ has derivative $1-3{\rho }^2$.

Step 2. The derivative vanishes at $\rho =\frac{1}{\sqrt{3}}\approx 0.577$, and $g$ increases before and decreases after, so this is the maximum.

Size. At the peak, $g=\frac{1}{\sqrt{3}}\cdot \frac{2}{3}\approx 0.385$, so the bias is about $-\frac{0.19}{n}$: for $n=20$, about $-0.01$.

Why it is biased at all. The sample covariance is unbiased, but the sample correlation divides it by the product of two sample standard deviations, a nonlinear operation. Unbiasedness does not survive nonlinear transformations, and here the result is pulled towards 0.