Near-Gaussian probability densities are common in many important physical applications. Here we develop an asymptotic expansion methodology for computing entropic functionals for such densities. The expansion proposed is a close relative of standard perturbation expansions in quantum field theory. We give novel results on the low-order effects of non-Gaussian even moments and asymmetry (e.g. skewness) on the entropy. The asymptotic expansion is also used to define a best fit maximum entropy density given a set of observed low order moments. The maximum entropy density estimation technique consists simply of the solution of a small set of algebraic equations and is therefore more straightforward numerically than classical maximum-entropy methods which rely on sophisticated convex optimization techniques.