Answer by Ronnie Pavlov for Prove or disprove: $R^{n+1} \supseteq R \cap R^2 \cap \cdots \cap R^n$ for every binary relation $R$ on a set of size $n$

domotorp's lovely solution is by far the best one, but here is an explicit counterexample for $n = 10$, I wonder if it's computationally tractable to figure out the max $n$ for which your statement holds.

$S = \{0,1,2,3,4,5,6,7,8,9\}$

$R = \{(0, 1), (0,6), (0,9), (1,2), (1,9), (2,3), (2,9), (3,4), (3,9), (4,5), (4,9), (5,1), (6,7), (6,9), (7,6)\}$

(It's not a typo that $8$ is not used.) This $R$ should have $(0,9) \in R \cap \dotsb \cap R^{10}$ but not $R^{11}$.