  We consider a one-dimensional classical Coulomb gas of $N$ like-charges in a
harmonic potential -- also known as the one-dimensional one-component plasma
(1dOCP). We compute analytically the probability distribution of the position
$x_{\max}$ of the rightmost charge in the limit of large $N$. We show that the
typical fluctuations of $x_{\max}$ around its mean are described by a
non-trivial scaling function, with asymmetric tails. This distribution is
different from the Tracy-Widom distribution of $x_{\max}$ for the Dyson's
log-gas. We also compute the large deviation functions of $x_{\max}$ explicitly
and show that the system exhibits a third-order phase transition, as in the
log-gas. Our theoretical predictions are verified numerically.
