what is the study called where the same people are tested repeatedly at different ages?

This? http://en.wikipedia.org/wiki/Cohort_study

Longitudinal

The study you are referring to is called a longitudinal study. In a longitudinal study, the same group of individuals is followed and tested over an extended period of time, often spanning several years or even decades. The goal of a longitudinal study is to observe changes and patterns in the variables being measured over time. By testing the same people repeatedly at different ages, researchers can gain valuable insights into the development, consequences, and effects of various factors or conditions.