We present broadband (radio, optical, and X-ray) light curves and spectra of the afterglows of four long-duration gamma-ray bursts (GRBs GRBs 090323, 090328, 090902B, and 090926A) detected by the Gamma-Ray Burst Monitor and Large Area Telescope (LAT) instruments on the Fermi satellite. With its wide spectral bandpass, extending to GeV energies, Fermi is sensitive to GRBs with very large isotropic energy releases (10$^54$ erg). Although rare, these events are particularly important for testing GRB central-engine models. When combined with spectroscopic redshifts, our afterglow data for these four events are able to constrain jet collimation angles, the density structure of the circumburst medium, and both the true radiated energy release and the kinetic energy of the outflows. In agreement with our earlier work, we find that the relativistic energy budget of at least one of these events (GRB 090926A) exceeds the canonical value of 10$^51$ erg by an order of magnitude. Such energies pose a severe challenge for models in which the GRB is powered by a magnetar or a neutrino-driven collapsar, but remain compatible with theoretical expectations for magnetohydrodynamical collapsar models (e.g., the Blandford- Znajek mechanism). Our jet opening angles (þeta) are similar to those found for pre-Fermi GRBs, but the large initial Lorentz factors (Γ$_0$) inferred from the detection of GeV photons imply þetaΓ$_0$ ≈ 70-90, values which are above those predicted in magnetohydrodynamic models of jet acceleration. Finally, we find that these Fermi-LAT events preferentially occur in a low-density circumburst environment, and we speculate that this might result from the lower mass-loss rates of their lower-metallicity progenitor stars. Future studies of Fermi-LAT afterglows at radio wavelengths with the order-of-magnitude improvement in sensitivity offered by the Extended Very Large Array should definitively establish the relativistic energy budgets of these events.