Is there a notion of "sequential" idempotence?
Your question would probably fit better on MathsStackExchange, but here is an answer.

I don't think there is any specific name for your definition. Since any semigroup is isomorphic to a transformation semigroup, your question is about finitely generated semigroups that are quotients of the semigroup $S_n$ with presentation
$$
\langle a_1, \ldots, a_n \mid a_1^2 = a_1, (a_1a_2)^2 = a_1a_2, \ldots, (a_1 \dotsm a_n)^2 = a_1 \dotsm a_n \rangle
$$
It turns out that $S_n$ is infinite for $n &gt; 1$, but I don't know whether it has been studied in the literature.