This paper addresses the state estimation problem of nonlinear systems. We formulate the problem using a minimum energy estimator (MEE) approach and propose an entropy penalized scheme to approximate the viscosity solution of the Hamilton-Jacobi equation that follows from the MEE formulation. We derive an explicit observer algorithm that is iterative and filtering-like, which continuously improves the state estimation as more measurements arise. In addition, we propose a computationally efficient procedure to estimate the state by performing an approximation of the nonlinear system along the trajectory of the estimate. In this case, for the first and second order approximations of the state equation, we derive a closed-form (iterative) solution for the Hessian of the entropy-like version of the optimal cost function of the MEE. We illustrate and contrast the performance of our algorithms with the extended Kalman filter (EKF) using specific nonlinear examples with the feature that the EKF do not converge to the correct value.