There are many, many examples in mathematics of operations $s$ satisfying $ss = s$ (i.e., idempotent operations).
Not quite as common, but still numerous, are operations $s$ satisfying $sss = s$, specifically, Galois connections from a poset to itself; see my recent post Abstract nonsense attribution and the articles referred to in replies to that post.
My (casual) question is, are there examples of important operations satisfying other relations of this kind, such as $sss = ss$? (This particular relation reminds me of the story of an article by X that ends with three footnotes: "The author thanks Y for translating the preceding article", "The author thanks Y for translating the preceding footnote", and "The author thanks Y for translating the preceding footnote"; infinite regress is avoided because, as X explained, "While I may not be able to translate a sentence I can certainly copy one!" Can anyone provide a web-reference for this? I think it's in Littlewood's Miscellany.)