Finite magma where the only equations are of the form "$t=t$"?

Question

Does there exist a finite set $S$ with a single binary operation $*$, where the only equational identities that hold are of the form $t=t$ for some term $t$?

Answer 1

In fact there is no finite magma which is generic even for one-variable terms.

Suppose $M$ is a finite magma. There are infinitely many one-variable terms but only finitely many functions $M\rightarrow M$; thus, there are distinct one-variable terms $t,s$ such that $t(a)=s(a)$ for all $a\in M$.

Note that it's important that we think about functions on $M$ as a whole rather than just individual elements: for example, for any set of $\vert M\vert+1$ one-variable terms $T$ there will be some $t,s\in T$ and $a\in M$ such that $t(a)=s(a)$, but there need not be distinct terms in $T$ which agree on all of $M$.

An interesting question is how many (parameter-free) one-variable terms there can be in a magma of size $n$ up to equality-of-corresponding-operations, as a function of $n$; of course we can get a trivial upper bound of $n^n$ by counting functions, but it's not clear to me that this upper bound is achievable. (We can also obviously ask the same question for arbitrary arity terms, with trivial upper bound of $n^{(n^{arity})}$, but already the unary case seems nontrivial to me.)