The successor function $S : \alpha \mapsto \alpha \cup \{\alpha\}$ where $\alpha$ is an ordinal is notable for being a strictly increasing function that has no fixed points, because it is discontinuous at limit ordinals since $\sup\{S(\beta)\mid\beta<\alpha\}=\alpha<S(\alpha)$
In ordinal notations, there are a variety of normal functions that are used as ordinal notations, such as $\epsilon_x,\phi(\alpha,0), '$ etc. However since they are all normal, they will eventually exhaust at fixed points by the normal function fixed point theorem.
This makes me wonder about generalisation of $S$ which has no fixed points, but made big jumps like e.g. the Veblen function, but while it is easy to define them, there seemed to be no systematic studies on constructing them.
Thus I am wondering if there are references, or otherwise out there that investigate the explicit constructions of generalised successors that are not simply of the form $+\beta$ for some ordinal $\beta$. Is there a name for these classes of non normal functions?