Hexagonal close-packed iron (hcp-Fe) is a main component of Earth's inner core. The difference in density between hcp-Fe and the inner core in the Preliminary Reference Earth Model (PREM) shows a density deficit, which implies an existence of light elements in the core. Sound velocities then provide an important constraint on the amount and kind of light elements in the core. Although seismological observations provide density-sound velocity data of Earth's core, there are few measurements in controlled laboratory conditions for comparison. We report the compressional sound velocity (VP) of hcp-Fe up to 163 GPa and 3000 K using inelastic X-ray scattering from a laser-heated sample in a diamond anvil cell. We propose a new high-temperature Birch's law for hcp-Fe, which gives us the VP of pure hcp-Fe up to core conditions. We find that Earth's inner core has a 4 to 5% smaller density and a 4 to 10% smaller VP than hcp-Fe. Our results demonstrate that components other than Fe in Earth's core are required to explain Earth's core density and velocity deficits compared to hcp-Fe. Assuming that the temperature effects on iron alloys are the same as those on hcp-Fe, we narrow down light elements in the inner core in terms of the velocity deficit. Hydrogen is a good candidate; thus, Earth's core may be a hidden hydrogen reservoir. Silicon and sulfur are also possible candidates and could show good agreement with PREM if we consider the presence of some melt in the inner core, anelasticity, and/or a premelting effect.