Answer
$u = \infty$
Work Step by Step
We can find the magnitude of the electric field as $r \to 0$:
$E = \frac{1}{4\pi~\epsilon_0}~\frac{\vert q \vert}{r^2}$
$E = \frac{1}{(4\pi)~(8.854\times 10^{-12}~C/V~m)}~\frac{1.6\times 10^{-19}~C}{r^2}$
$E = \infty$
We can find the energy density:
$u = \frac{1}{2}~\epsilon_0~E^2$
$u = \frac{1}{2}~(8.854\times 10^{-12}~C/V~m)~E^2$
$u = \infty$
