Consider an irrational rotation of a unit circle and a real continuous function. A point is declared ‘maximizing’ if the growth of the ergodic sums at this point is maximal up to an additive constant. In the case of two-sided ergodic sums, the existence of a maximizing point for a continuous function implies that it is the coboundary of a continuous function. In contrast, we build, for the ‘usual’ one-sided ergodic sums, examples in Hölder or smooth classes, indicating that all kinds of behaviour of the function with respect to the dynamical system are possible. We also show that generic continuous functions are without maximizing points, not only for rotations, but also for the transformation 2x mod 1. For this latter transformation, it is known that any Hölder continuous function has a maximizing point.