A finite semigroup $S$ has finite essential arity iff it is locally trivial, that is, $eSe=e$ for all idempotents $e\in S$.
Pf. First suppose that $S$ is not locally trivial. The either $S$ contains a subsemigroup isomorphic to the 2-element semilattice $\lbrace 0,1\rbrace$ with multiplication or it contains a non-trivial group $G$ as a subsemigroup.
In the first case, the variety generated by $S$ contains the free semilattice on any set (i.e. the power set under union with singletons as free generators) and so the term function given by a word on m-letters depends on all those letters. If $S$ contains a non-trivial group $G$, then it contains a cyclic group of prime order $p$ and hence vector spaces over $\mathbb F_p$. But then again the term function depending on any word in m-letters depends on all letters by considering a vector space of dimension m.
So finite essential arity implies locally trivial.
Conversely, if $S$ is locally trivial, then it is known that $S$ satisfies an identity of the form $x_1\cdots x_myz_1\cdots z_n = x_1\cdots x_mz_1\cdots z_n$ and so any term operation coming from a word depends only on at most $m+n$ variables (the prefix of length $m$ and the suffix of length $n$).
In particular, no non-trivial monoid has finite essential arity.