We consider misaligned functional data, where data registration is necessary for proper statistical analysis. This paper proposes to treat misalignment as a nonlinear random effect, which makes simultaneous likelihood inference for horizontal and vertical effects possible. By simultaneously fitting the model and registering data, the proposed method estimates parameters and predicts random effects more precisely than conventional methods that register data in preprocessing. The ability of the model to estimate both hyperparameters and predict horizontal and vertical effects are illustrated on both simulated and real data.