The state estimation of repetitive processes with periodically repeated trajectories can be interpreted as the dual task of iterative learning control design. While the latter has been widely investigated over the last two decades, only few approaches exist for the design of iterative learning observers. However, the exploitation of the knowledge about periodically repeated trajectories, which occur among others in pick and place tasks in robotics as well as in charging and discharging of batteries, offers the opportunity to enhance the estimation accuracy from one execution of the control task to the next. In this paper, we generalize a linear stochastic approach for iterative learning state estimation, inspired by the Kalman filter in terms of a minimization of the estimation error covariance, to the class of models with bounded parameter uncertainty and to nonlinear ones that can be represented by means of quasi-linear discrete-time state-space representations. To solve this task, a novel combination of set-valued ellipsoidal state enclosure techniques with the aforementioned stochastic iterative learning state estimator is presented and visualized for a quasi-linear model of the charging/discharging dynamics of Lithium-Ion batteries.