Do (co)density (co)monadic constructions stablize? Under good conditions [1], any functor $F: C \to D$ induces a codensity monad $T: D \to D$ as a right Kan extension of $F$ along itself. It does not say explicitly, but by considering left/right Kan extension/lift, it could make sense to talk about four variants: (co)density (co)monads. These subsume many natural mathematical constructions [1].
Taking composites, e.g. $TF: C \to D$, we get four natural new functors of the same type for free! And since $TF$ is a functor, we can construct its (co)density (co)monads iteratively.
Question

*

*Does such iterative construction stablize?

*If not, what's the state of the art about it?

Reference

*

*[1] https://golem.ph.utexas.edu/category/2012/09/where_do_monads_come_from.html
 A: Not sure about state of the art, but here are a couple of comments.
As is pointed out in your reference, if $F$ has a left adjoint $G$, then $T=FG$. If $F$ happens to be fully faithful, then $G$ is a localization functor. In this case $TF\cong F$, so the construction stabilizes. This holds regardless of whether $F$ has a left adjoint: if $F$ is fully faithful embedding then $TF=F$.
But if $F$ is just a functor with a left adjoint, there is no reason for the construction to stabilize. What happens is that $F$ induces a functor from $C$ to the category of $T$-algebras (which is usually denoted by the same letter $F$). The functor $F$ is called monadic if this induced functor is an equivalence of categories. Beck's monadicity theorem gives a criterion that characterizes monadic functors.
Perhaps more relevant to "state of the art", in the last decade there has been an interest in homotopic analogues of this set-up, e.g. in the context of Quillen model categories or $\infty$-categories. You can read more about it for example in the chapter Monads and the Barr-Beck Theorem in Lurie's Higher Algebra. I also recommend these notes on homotopic descent.
