Laver tables of order $n$ provide a potential answer (although it's not proven to be independent of ZFC, and may not be independent of ZFC).

The *Laver table* of order $n$ is the $2^n \times 2^n$ table of an operation $\star$ defined recursively by
$$
p \star 1 = p+1 \bmod {2^n}\\
p \star (q \star r) = (p \star q) \star (p \star r).
$$
They arise naturally in the study of (self) left-distributive systems (systems satisfying the second axiom above).

The first row of this table ($1 \star p$) is always periodic with some period $m$ dividing $2^n$. Let $p(n)$ be this periodicity. Assuming an extremely large cardinal, Laver showed that $p(n)$ increases without bound as $n$ increases. (The consistency of this large cardinal axiom, the existence of a self-embedded cardinal, is apparently in doubt.) Under this same large cardinal hypothesis, Dougherty showed, for instance, that the first $n$ for which $p(n)>m$ grows faster than any primitive recursive function of $m$.

I'm not sure what the current state of belief is on whether this statement is independent of ZFC. There were various other statements first proved by Laver using this large cardinal hypothesis that were later proved by Dehornoy; for instance, there is an algorithm to decide the word problem in a free left-distributive algebra.

I could have sworn there was a statement in this theory that was known to be independent of ZFC, but I couldn't find it when I looked.

I'm not a logician. Apologies if I'm misstating any results.

References:

https://googology.fandom.com/wiki/Laver_table

Randall Dougherty, *Critical points in an Algebra of Elementary Embeddings*, Ann. Pure Appl. Logic **65** (1993), no. 3, 211-241, http://arxiv.org/abs/math/9205202