User blog comment:KthulhuHimself/Ordinal Arithmetic Notation (OAN)/@comment-35470197-20191112141154

I have seen definitions of transfinite hyperoperators several times. I think that they are helpful also for you. For example, Hexirp's formulations are available here, and their analyses are avaiable here. (The blog posts are written in Japanese, but the definitions are written in formulae.)