We study asymptotics for orthogonal polynomials and other extremal polynomials on infinite discrete sets, typical examples being the Meixner polynomials and the Charlier polynomials. Following ideas of Rakhmanov, Dragnev and Saff, weshow that the asymptotic behaviour is governed by a constrained extremal energy problem for logarithmic potentials, which can be solved explicitly. We give formulas for the contracted zero distributions, the $n$th root asymptotics and the asymptotics of the largest zeros.