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This paper proposes adaptive iterative learning control (ILC) schemes for continuous-time parametric nonlinear systems with iteration lengths that randomly vary. As opposed to the existing ILC works that feature nonuniform trial lengths, this paper is applicable to nonlinear systems that do not satisfy the globally Lipschitz continuous condition. In addition, this paper introduces a novel composite energy function based on newly defined virtual tracking error information for proving the asymptotical convergence. Both an original update algorithm and a projection-based update algorithm for estimating the unknown parameters are proposed. Extensions to cases with unknown input gains, iteration-varying tracking references, nonparametric uncertainty, high-order nonlinear systems, and multi-input-multi-output systems are all elaborated upon. Illustrative simulations are provided to verify the theoretical results.