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](https://mathoverflow.net/questions/53022/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.)



singly-generated monoids in mathematics
Think about the sheaves on some site as a full subcategory of presheaves: $Sh(C)\to PSh(C)$. This has a left adjoint, called sheafification. There are various ways to construct the sheafification, but one of them uses something called the plus-construction. For any presheaf $F$ it gives an associated presheaf $F^+$, which is always separated but may not be a sheaf. (Being separated means that there is at most one way to glue local bits of data, as opposed to exactly one way for a sheaf.)


If you do this a second time, to get $F^{++}$, you do get a sheaf, and if you do it a third time it has no further effect. Thus $sss=ss$ if $s$ is this plus-construction. 


There is a higher-dimensional version involving Cat-enriched presheaves and stacks, where you have to do the analogous construction three times. This would give an example of $ssss=sss$. 


I don't know if this continues in still higher dimensions. 


