Let $M_d$ be the maximum of a standard Bessel bridge of dimension $d$. A series formula for $P(M_d < a)$ due to Gikhman and Kiefer for $d = 1,2, \ldots$ is shown to be valid for all real $d >0$. Various other characterizations of the distribution of $M_d$ are given, including formulae for its Mellin transform, which is an entire function. The asymptotic distribution of $M_d$ as is described both as $d$ tends to $\infty$ and as $d$ tends to $0$. Keywords: Brownian bridge, Brownian excursion, Brownian scaling, local time, Bessel process, zeros of Bessel functions, Riemann zeta function.