This paper presents a method for recursively estimating the static parameters of linear or nonlinear stochastic dynamic systems given the systems' inputs and outputs. The paper accomplishes this objective by combining polynomial chaos theory with maximum likelihood estimation. The parameter estimates are calculated in a recursive or iterative manner. To the best of the author's knowledge, this is the first paper to address recursive maximum likelihood parameter estimation using polynomial chaos theory. The proposed approach is demonstrated on two systems: a linear 2nd order system with unknown damping and natural frequency, and a nonlinear Van der Pol oscillator with an unknown nonlinear damping coefficient. Because this recursive estimator is applicable to nonlinear systems, the authors portend that this novel formulation will be useful for a broad range of estimation problems.