We use exogenous variation in the skills that children have at the beginning of kindergarten to measure the extent to which “skills beget skills”. Children who are relatively older when they begin kindergarten score higher on measures of cognitive and non-cognitive achievement at the beginning of kindergarten. Their scores on cognitive assessments grow faster during kindergarten and first grade, consistent with complementarities between existing stocks of skills and the acquisition of additional skills. However, after first grade the scores of younger entrants catch up. We show that this catch up is due to the influence of schools. After controlling for this influence, we observe a net increase in the skill gap between older and younger school entrants between kindergarten and eighth grade, although growth in the skill gap is not consistent over time.