Summary: This paper addresses the problem of adaptive neural tracking control for a class of nonstrict-feedback stochastic nonlinear time-delayed systems with unknown input saturation and unknown disturbance input. A smooth function is selected to approximate the non-smooth saturation function. By using the command filter method scheme, the well-known problem of the explosion of complexity, which appears in the classical backstepping methods, is avoided. To approximate the unknown nonlinear functions, the radial basis function neural networks (NNs) are deployed. In addition, considering the minimal learning parameter method makes the updating law independent of the number of neural nodes and the order of the system. By using the mean-value theorem, the adaptive NN tracking control law is obtained systematically regardless of pre-known knowledge of input saturation bounds. The proposed control scheme guarantees that the state trajectories of the closed-loop system are semi-global uniformly ultimately bounded in probability and the tracking error finally stays in a small neighborhood around the origin. Eventually, numerical and practical examples show the performance of the proposed controller design.