We consider the process of excitation of a strong axisymmetrical plasma wave by a relativistic electron bunch. It is well known that in one-dimensional geometry the excited wave becomes essentially nonlinear when the beam density approaches half the plasma density, and wavebreaking occurs at beam densities higher than that. Such a simple relation, however, does not hold in three dimensions. We find that the total beam current, not only the current density, should exceed a certain threshold for nonlinearity to become visible. Just recently, new developments in accelerator technology have made it possible to produce relativistic electron beams powerful enough to enter this nonlinear regime. We show that a realistic electron bunch from an advanced accelerator can generate in its wake accelerating fields well above 10 GeV m-1.