In this paper we study the adaptive tracking control for nonlinear ARX (NARX) systems with additive observation noise. Compared to the existing works the dynamical function in the model is allowed to process arbitrary growth rate with respect to the input argument. The adaptive control is designed on the basis of stochastic approximation algorithm with expanding truncations (SAAWET), and it is pointed out that there is a trade-off between the growth rate of the dynamical function with respect to control argument and that of the truncation bound to be chosen. We proved that the tracking purpose is achieved and the numerical simulation given in the paper justifies the theoretical assertions.