  We consider a semilinear parabolic equation with a large class of
nonlinearities without any growth conditions. We discretize the problem with a
discontinuous Galerkin scheme dG(0) in time (which is a variant of the implicit
Euler scheme) and with conforming finite elements in space. The main
contribution of this paper is the proof of the uniform boundedness of the
discrete solution. This allows us to obtain optimal error estimates with
respect to various norms.
