I know for example that the first level of the Grzegorczyk hierarchy contains the functions which enumerate the c.e sets and that it has an interesting relation to the provably total functions in certain axiomatizations of arithmetic due to work of Kriesel, Girard and others. I was wondering if we know of some other interesing connections. For example

Can we say anything about the hierarchy levels and escape or domination properties of the functions enumerated by functions from said levels? I am looking for any connections like this from the literature.