Hydrated electrons are anionic species that are formed when an excess electron is introduced into liquid water. Building an understanding of how hydrated electrons behave in solution has been a long-standing effort of simulation methods, of which density functional theory (DFT) has come to the fore in recent years. The ability of DFT to model the reactive chemistry of hydrated electrons is an attractive advantage over semi-classical methodologies; however, relatively few density functional approximations (DFAs) have been used for the hydrated electron simulations presented in the literature. Here, we simulate hydrated electron systems using a series of exchange-correlation (XC) functionals spanning Jacob's Ladder. We calculate a variety of experimental and other observables of the hydrated electron and compare the functional dependence for each quantity. We find that the formation of a stable, localized hydrated electron is not necessarily limited to hybrid XC functionals, and that some hybrid functionals produce delocalized hydrated electrons or even unphysically react with the surrounding water. We further characterize how different DFAs impact the solvent structure and predicted spectroscopy of the hydrated electron, considering several methods for calculating the hydrated electron's absorption spectrum for best comparison between structures generated using different density functionals. None of the dozen or so DFAs that we investigated are able to correctly predict the hydrated electron's spectroscopy, vertical detachment energy or molar solvation volume.