An adaptive iterative learning control method is proposed for a class of nonlinear strict-feedback discrete-time systems with random initial conditions and iteration-varying desired trajectories. An n-step ahead predictor approach is employed to estimate the future states in the control design. Discrete Nussbaum gain method is utilised to deal with the lack of a priori knowledge of control directions. The proposed control algorithm guarantees the boundedness of all the signals in the controlled system. The tracking error converges to zero asymptotically along the iterative learning axis except for beginning states affected by random initial conditions. The effectiveness of the proposed control scheme is verified through numerical simulation.