Here is a list of some of the most important papers and books somewhat related to classical Laver tables. I am still in the process of adding more references and giving descriptions of the references.
Handbook of Set Theory (Chapter 11, 2010)-In this chapter of the Handbook of set theory, Patrick Dehornoy gives an exposition of the self-distributive structures that arise from rank-into-rank embeddings. This exposition is essentially the same as Dehornoy’s exposition of algebras of elementary embeddings in his book Braids and Self-distributivity.
Braids and Self-Distributivity(Patrick Dehornoy, 2000)-This textbook is the primary reference for self-distributive algebras regarding topics such as their relation to braid groups, freeness and term algebras, classical Laver tables, and algebras of elementary embeddings. The chapter on algebras of elementary embeddings (chapter 12) is accessible to non-set theorists since it provides the required background material on set theory.
Quandles: An Introduction to the Algebra of Knots(Mohamed Elhamdadi and Sam Nelson, 2015)-Quandles are the idempotent self-distributive algebraic structures where every inner endomorphism is an automorphism. Quandles are used in knot theory since the knot quandle of a knot characterizes the knot up to isomorphism. However, Laver tables and quandles are vastly different classes of self-distributive structures.
Critical points in an algebra of elementary embeddings I(1992, Randall Dougherty)
Critical points in an algebra of elementary embeddings II(1995, Randall Dougherty)
On the Algebra of Elementary Embeddings of a Rank into Itself(1995, Richard Laver)
The left distributive law and the freeness of an algebra of elementary embeddings(1992, Richard Laver)
Finite left-distributive algebras and embedding algebras(Thomas Jech and Randall Dougherty, 1997)
Papers after 2010
Two- and three-cocycles for Laver tables(Patrick Dehornoy and Victoria Lebed, 2014)
Laver’s result and low-dimensional topology(Patrick Dehornoy, 2015)