Allotropes of carbon, such as diamond and graphene, are among the best conductors of heat. We monitored the evolution of thermal conductivity in thin graphite as a function of temperature and thickness and found an intimate link between high conductivity, thickness, and phonon hydrodynamics. The room-temperature in-plane thermal conductivity of 8.5-micrometer-thick graphite was 4300 watts per meter-kelvina value well above that for diamond and slightly larger than in isotopically purified graphene. Warming enhances thermal diffusivity across a wide temperature range, supporting partially hydrodynamic phonon flow. The enhancement of thermal conductivity that we observed with decreasing thickness points to a correlation between the out-of-plane momentum of phonons and the fraction of momentum-relaxing collisions. We argue that this is due to the extreme phonon dispersion anisotropy in graphite.