We consider random Schr\"odinger operators of the form \(\Delta+\xi\), where \(\Delta\) is the lattice Laplacian on \(\mathbb Z^d\) and \(\xi\) is an i.i.d. random field, and study the extreme order statistics of the eigenvalues for this operator restricted to large but finite subsets of \(\mathbb Z^d\). We show that for \(\xi\) with a doubly-exponential type of upper tail, the upper extreme order statistics of the eigenvalues falls into the Gumbel max-order class. The corresponding eigenfunctions are exponentially localized in regions where \(\xi\) takes large, and properly arranged, values. A new and self-contained argument is thus provided for Anderson localization at the spectral edge which permits a rather explicit description of the shape of the potential and the eigenfunctions. Our study serves as an input into the analysis of an associated parabolic Anderson problem.