Output regulation for a discrete-time nonlinear plant driven by an exogenous signal is addressed. In the absence of the standard exosystem hypothesis, we assume availability of the current and l past values of the exogenous signal for use in the control law. We formulate an approximate output regulation problem and characterize solvability in terms of the existence of a certain invariant manifold. The control law yields a closed-loop system with output exhibiting ultimate boundedness properties with respect to the (l + 1) difference of the exogenous signal. A perturbation analysis of robustness properties is also provided.