We present Chandra, XMM-Newton, and Swift observations of the quiescent neutron star in the transient low-mass X-ray binary MAXI J0556-332. Observations of the source made during outburst (with the Rossi X-ray Timing Explorer) reveal tracks in its X-ray color-color and hardness-intensity diagrams that closely resemble those of the neutron-star Z sources, suggesting that MAXI J0556-332 had near- or super-Eddington luminosities for a large part of its ~16 month outburst. A comparison of these diagrams with those of other Z sources suggests a source distance of 46 ± 15 kpc. Fits to the quiescent spectra of MAXI J0556-332 with a neutron-star atmosphere model (with or without a power-law component) result in distance estimates of 45 ± 3 kpc, for a neutron-star radius of 10 km and a mass of 1.4 M ☉. The spectra show the effective surface temperature of the neutron star decreasing monotonically over the first ~500 days of quiescence, except for two observations that were likely affected by enhanced low-level accretion. The temperatures we obtain for the fits that include a power law (kT_eff∞ = 184-308 eV) are much higher than those seen for any other neutron star heated by accretion, while the inferred cooling (e-folding) timescale (~200 days) is similar to other sources. Fits without a power law yield higher temperatures (kT_eff∞ = 190-336 eV) and a shorter e-folding time (~160 days). Our results suggest that the heating of the neutron-star crust in MAXI J0556-332 was considerably more efficient than for other systems, possibly indicating additional or more efficient shallow heat sources in its crust.