For each $n$, there is a finite left self distributive algebra on one generator with $2^n$ elements. These are known as the Laver tables and have a purely number-theoretic definition (although see my other post concerning computation in them). These naturally project down to the smaller tables in such a way to give rise to an inverse limit. In this inverse limit, there is a copy of the generator and one can use it to generate an algebra $A_\infty$.
Here is the interesting part: the freeness of $A_\infty$ has been established from the assumption that, for each $n$, there is an $n$-huge cardinal (these are among the strongest of the large cardinal hypotheses). No proof is known in ZFC. See Laver's articles in Advances in Mathematics or Dehornoy's book "Braids and self distributivity for more information. In my opinion this is the greatest problem in reverse mathematics.
These tables did originate in set theory, but they have natural connections to the study of braids (Dehornoy's book is testimony to this).