This paper considers the problem of state observation for nonlinear dynamics. While model-based observer synthesis is difficult due to the need of solving partial differential equations, this work proposes an efficient model-free, data-driven approach based on online learning. Specifically, by considering the observer dynamics as a Chen-Fliess series, the estimation of its coefficients has a least squares formulation. Since the series converges only locally, the coefficients are recursively updated, resulting in an online optimization scheme driven by instantaneous gradients. When the state trajectories are available, the online least squares guarantees an ultimate upper bound of average observation error proportional to the average variation of states. In the realistic situations where the states cannot be measured, the immersed linear dynamics based on the Kazantzis-Kravaris/Luenberger structure is assigned, followed by online kernel principal component analysis for dimensionality reduction. The proposed approach is demonstrated by a limit cycle dynamics and a chaotic system.