big picture - Various concepts of "closure" or "completion" in mathematics

Ah! You edited your question! I had to delete my answer. Anyway, here is a general scenario where idempotent operations such as the one you want arise:

In this para I am going to be vague. But the examples below given should illustrate what I have in mind.Ok, so, You have "some structure" somewhere. You want to go to the "maximal" of such a thing. You have a natural ordering on such structures you want. And also it so happens that the union of a chain of such stuff is again such a thing. Then you apply Zorn's lemma to find the maximal thing. And this operation of going and finding the maximal thing is an "idempotent completion" in your sense.

There are plenty of examples. A few:

$1$. A set of linearly independent vectors in a vector space is enlarged to a basis.

$2$. An algebraic extension of a field is enlarged to the algebraic closure.

$3$. A separable extension of a field, is enlarged to separable closure.

$4$. A differentiable atlas on a smooth manifold is enlarged to a maximal one, ie., a differentiable structure.

$5$. A certain functional on a Banach space is enlarged to fill the whole space, as in the proof of Hahn-Banach theorem.

And so on, nearly in fact every application of Zorn's lemma.

This is not a functorial way to go; but the construction as an operation is idempotent. And in some way, such as in the construction of the algebraic closure, we have an isomorphism of two such different constructions.