I think the comment by @James Martin answers my question.
The number of clusters equals the number of pairs of points that are mutually nearest neighbors. The probability that a point is in a mutually nearest neighbor pair is $p=\dfrac{1}{\frac43+\frac{\sqrt3}{2\pi}}\approx0.62150$ (proof). So for $n$ points, the expected number of pairs that are mutually nearest neighbors approaches $\frac{np}{2}$. So the expected number of points per cluster approaches $\frac2p=\frac83+\frac{\sqrt3}{\pi}\approx 3.218$.
(I may not be using terminology precisely enough; feel free to edit.)