This paper is a contribution to the study of hereditary classes of finite graphs. We classify these classes according to the number of prime structures they contain. We consider such classes that are \emph{minimal prime}: classes that contain infinitely many primes but every proper hereditary subclass contains only finitely many primes. We give a complete characterization of such classes. In fact, each one of these classes is a well quasi ordered age and there are uncountably many of them. Eleven of these ages remain well quasi ordered when labels in a well quasi ordering are added. Among the remaining ones, countably many remain well quasi ordered when one label is added. Except for six examples, members of these ages we characterize are permutation graphs. In fact, every age which is not among the eleven ones is the age of a graph associated to a uniformly recurrent $0$-$1$ word on the integers. A characterization of minimal prime classes of posets and bichains is also provided.