Forgive me if this question is not well thought out. I don't know how else to ask it.

The nlab page on completion gives some examples of completions which are left adjoints. These completions are "free" and describe the most effortless way of getting particular properties. Particularly, these left adjoint feel as though they don't really "look inside" the object to which they're applied.

On the other hand, consider the coskeleton functor $$\mathrm{coskel}_n:s\mathsf{Set}_{\leq n-1}\longrightarrow s\mathsf{Set}_{\leq n} $$ defined by "filling in" $n$-dimensional simplicial holes. This is also a process of "completion", but it is the opposite of effortless: it seems like the most laborious way to get an $n$-truncated simplicial set from an $(n-1)$-truncated one, involving a difficult "search for simplicial holes" throughout the $(n-1)^\text{th}$ level.

Thus the adjoint triple $\mathrm{skel}_n\dashv \mathrm{tr}_n\dashv\mathrm{coskel}_n$ describes a left adjoint to truncation which is truly effortless, and a right adjoint which is maximally laborious.

What are some more examples of right adjoints which deserve to be called "laborious completions"? Perhaps such right adjoint completions only arise as the rightmost adjoint in a triple, so I'd be especially interested in such examples.