This paper discusses the existence of an asymptotic expansion for the global error of the implicit Euler scheme applied to stiff nonlinear systems of ordinary differential equations. It is shown that in strongly stiff situations, a full asymptotic expansion exists at all gridpoints. For the mildly stiff case it is shown that the full order of the remainder term, which inevitably breaks down at the first gridpoints after a stepsize change, reappears at the subsequent gridpoints. Our analysis is based on singular perturbation techniques.