This paper investigates the growth over time of the size of higher education institutions (HEIs), as measured by the number of academic staff, and its association with HEI and country attributes. We analyze a sample of 837 HEIs from 18 countries derived from the European Tertiary Education Register (ETER) and from the European Micro Data dataset (EUMIDA) for the years 2008 and 2012. Our analysis shows that (1) HEIs growth is largely proportional to their size, leading to a nearly log-normal distribution of size (Gibrat's law), even if small institutions tend to grow faster; (2) the growth of the number of students and HEI's reputation level positively influences HEI growth. Consequently (3) small HEIs need a lower level of reputation and less growth of students to continue growing over time, while only highly reputed HEIs are able to maintain a large size over time. Our results are relevant to understand the extent to which cumulative effects lead to a lasting concentration of resources in the HE system and whether public policies are able to redistribute resources based on merit.