Progressive crystallisation of Earth's inner core over geological times drives convection in the outer core and the generation of the Earth’s magnetic field. Resolving the rate and pattern of inner core growth is thus crucial to understanding the evolution of the geodynamo. The growth history of Earth’s inner core is likely recorded in the distribution and strength of seismic anisotropy arising from deformation texturing constrained by boundary conditions at the inner-core solid-fluid boundary. Travel times of seismic body waves indicate that seismic anisotropy increases with depth. Here we find that the strongest anisotropy is offset from Earth's rotation axis. Using geodynamic growth models and mineral physics calculations, we simulate the development of inner core anisotropy in a self-consistent manner. We show for the first time that an inner core model composed of hexagonally close-packed iron-nickel alloy, deformed by a combination of preferential equatorial growth and slow translation can match the seismic observations without requiring the introduction of hemispheres with sharp boundaries. We find a model of the inner core growth history compatible with external constraints from outer core dynamics, supporting arguments for a relatively young inner core (~0.5-1.5 Ga) and a viscosity >1018 Pa-s.