X-ray photoelectron spectra are typically broadened by the inclusion of contributions from the exciting X radiation, as well as the spectrometer itself. A procedure for removal of much of this broadening using a version of the maximum entropy method is described. This method uses a nonlinear functional (the spectral "entropy" or information content) to stabilize the noise amplification problem inherent to deconvolution of real data. A less ambiguous solution is therefore obtained compared to previous approaches which use linear estimates. This algorithm avoids the subjective nature of many previous deconvolution methods by assuming that the informational uncertainty is always maximized within the constraint of the data provided. This large scale, nonlinear optimization problem can be solved on a fast personal computer using a sequential quadratic programming algorithm. For spectra with adequately high signal/noise, the linewidths produced approach the limiting core hole lifetime values. The method is applied to study the extended growth of an oxide layer on clean polycrystalline aluminum surfaces following exposure to water vapor. For the early growth stages, the spectral detail is comparable to recent synchrotron-based studies where one or more precursor phases are detected at the interface prior to the nucleation of the extended oxide film. In a related study of the oxidation of gold-aluminum alloy surface films, the evolution of the gold-aluminum near-surface phase could be followed through changes in the Au(4f) and the Al(2p) spectra. From the latter, the surface oxides grown on the alloy surface are seen to differ distinctly from those on pure aluminum.