Laver showed in 1995 that the period of the first row of certain [Laver tables](https://en.wikipedia.org/wiki/Laver_table) is unbounded, assuming that a rank-into-rank cardinal exists.


The most accessible proof of his result that I was able to find is in chapter 12 of Patrick Dehornoy's Braids and Self-Distributivity ([Springer 2000](https://link.springer.com/book/10.1007/978-3-0348-8442-6)). The proof is quite technical. Laver defines an algebra on the models of set-theory - technically, on certain elementary embeddings of huge sets. He defines two operations on such elementary embeddings, quotients out certain large infinities to get finite results, and investigates their properties.


My question is: why should elementary embeddings of ZFC have anything to do with the periodicity of these finite tables?


More generally, is there some guiding intuition I can use to make sense of Laver's very complex construction? I find it difficult to understand how Laver could have plowed through this weird and intricate technical construction without having some reason to think it would lead to some kind of specific finitary result.



Motivation for Laver's use of large cardinals to show finite combinatorial properties of Laver tables
As was already mentioned in the comments, the premise of the question is somewhat backwards. Indeed, looking at [Laver's paper](https://www.sciencedirect.com/science/article/pii/S0001870885710146?via%3Dihub), the combinatorial structures now known as Laver tables were not at all his initial motivation. Instead, from the start and throughout Laver was interested in elementary embeddings of some $V\_\lambda$ and the algebras they produce under certain operations, one of which is "applying" one elementary embedding to another. This operation satisfies a relation $a(bc)=(ab)(ac)$ (and thus defines what is sometimes known as a [shelf](https://ncatlab.org/nlab/show/shelf)). For instance, Laver has shown that the shelf generated by one elementary embedding $j$, which he denotes $\mathscr A\_j$, is free.


Now, $\mathscr A\_j$ is a large and complicated algebra. One of Laver's results (essentially Theorem 11 of the paper) is that the study of $\mathscr A\_j$ can be reduced to studying certain finite algebras, $\mathscr A\_{j,n}$, which have $2^n$ elements. These algebras are again defined using elementary embeddings, but as Laver explains after the theorem, these algebras also have a completely explicit description not referring to any large cardinals - these are the Laver tables that you are familiar with.


Therefore Laver's motivation was studying certain algebras which arise naturally if you are interested in large cardinals. Any relation to finitary structures was an afterthought to him.


