Minimum Energy Estimation is a way of filtering the state of a nonlinear system from partial and inexact measurements. It is a generalization of Gauss' method of least squares. Its application to filtering of control systems goes back at least to Mortenson who called it Maximum Likelyhood Estimation. For linear, Gaussian systems it reduces to maximum likelihood estimation (aka Kalman Filtering) but this is not true for nonlinear systems. We prefer the name Minimum Energy Estimation (MEE) that was introduced by Hijab. Both Mortenson and Hijab dealt with systems in continuous time, we extend their methods to discrete time systems and show how Taylor polynomial techniques can lessen the computational burden. The degree one version is equivalent to the Extended Kalman Filter in Information form. We apply this and the degree three version to problem of estimating the state of the three dimensional chaotic Lorenz Attractor from a one dimensional measurement. They both perform well but we find that the degree three version is 40% more accurate on average.