We describe the direct observation of entropy-driven crystallization in simulations of dense packings of linear hard-sphere chains. Crystal nuclei form spontaneously in the phase coexistence region independently of chain length. Incipient nuclei consistently develop well defined, stack-faulted layered morphologies with a single stacking direction. These morphologies deviate markedly from those of monomeric analogs. The ordering transition is driven by the increase in translational entropy: ordered sites exhibit enhanced mobility as their local free volume becomes more spherical and symmetric.