An adaptive iterative learning control scheme is proposed for a class of discrete-time nonlinear systems with random initial conditions and iteration-varying desired trajectories. The discrete Nussbaum gain method is incorporated into the control design to tackle the problem associated with the lack of a priori knowledge of the 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. The effectiveness of the proposed control law is verified through numerical simulation.