We will explain how to associate a Newton polygon to an element in GL(n) over certain fields of formal Laurent series. If we refine this construction by restricting to Newton polygons coming from elements in a fixed stratum of the affine Bruhat decomposition, there will be a unique maximum Newton polygon. Finding this maximal Newton polygon can be phrased in terms of the Bruhat order on the affine symmetric group, which then leads to a closed formula given in terms of directed paths in the quantum Bruhat graph. This graph was introduced by Brenti, Fomin, and Postnikov, and has appeared in the context of computing monomials in the quantum cohomology of the Grassmannian. Interestingly, the formula for the minimal monomial in the product of two Schubert classes in QH(G/B) coincides with that for the maximal Newton polygon, so we will conclude by mentioning subsequent open questions in both the Newton polygon and quantum cohomology contexts.