Which axiom can almost determine the magma with one element?

Question

The axiom $((a * b) * c) * (a * ((a * c) * a)) = c$ uniquely determines Boolean algebra, an example of a single axiom giving a magma an "interesting" structure. What is the fewest number of times that * can occur in an axiom for a magma while still only allowing the empty magma and the magma with one element and a magma with $2$ elements?

Answer 1

Per the comments, the OP is looking for an equational axiom. No such equation exists, however, due to the following fact:

If $\mathcal{A},\mathcal{B}$ are each models of an equation $\mathsf{E}$, then the direct product $\mathcal{A}\times\mathcal{B}$ is also a model of $\mathsf{E}$.

In particular, if $\mathsf{E}$ has a model of size $2$, then it has models of size $2^k$ for every $k$. So no equation can allow a model with two elements but not allow models with more elements.