Veblen $\phi$ functions extend the $\xi \mapsto \phi(\xi) := \omega^\xi$ and the $\xi \mapsto \phi(1,\xi) := \varepsilon_\xi$ functions on the ordinals by repeatedly taking fixed points (I won't repeat the definitions, which are on Wikipedia). For a reason that isn't entirely clear to me, Veblen functions of just *two* variables seem to have gotten all the attention (to the point that the range $\phi(1,0,0)$ of the notation system they provide has a special name: the "Feferman-Schütte ordinal").

Essentially the only text I was able to find which discusses the Veblen functions of more than two variables is Veblen's original paper (published in 1908, "Continuous Increasing Functions of Finite and Transfinite Ordinals"), which is difficult to read: he starts his ordinals at $1$ and uses slightly different conventions on the $\phi$ functions than are now more common (e.g., on the order of the variables), and so on; more importantly, he does not discuss normal forms for ordinals produced by these functions.

So, is there a more modern account that I missed? Ideally one that would describe in some detail the system of ordinal notations that can be extracted from Veblen functions of $\omega$ variables (the "small" Veblen ordinal) or of the smallest fixed point of $\xi \mapsto \phi(1^{(\xi)},\ldots,0)$ (the "large" Veblen ordinal). But even a modern "translation" of Veblen's paper would be of some help.

**Edit:** There seems to be some kind of ordinal notation system related to Veblen functions of *three* variables (I think) in Ackermann's 1951 paper "Konstruktiver Aufbau eines Abschnitts der zweiten Cantorschen Zahlenklasse", though the notations are also very old (and the ordinals also start at $1$, apparently zero still hadn't been invented in 1951). I'll try to read it and see whether it's interesting.