Iterating Monad-Comonads structures Let $(T, \mu ,  \eta  )$ a monad on  the category  $\mathscr{C}$ , with the usual EM (Eilenberg-Moore) adjunction $\langle F_T, U_T, \eta_Y, \epsilon_T \rangle:  \mathscr{C}^T \to  \mathscr{C}$  where $\mathscr{C}^T$ is the category of $T$-agebras, objects are couples  $(A, a)$ where $a: T(A) \to  A$ such that $a \circ  \eta_A=1$, $a \circ  \mu= a \circ  T(a)  $.
The above adjunction induce on $\mathscr{C}$  the some triple $T$, but on $\mathscr{C}^T$  there is a cotriple $(cT, \delta, \epsilon )$ with $cT(A, a)= (T(A), \mu_A)$, $\epsilon_{(A, a)}: cT(A, a) \to  (A, a)$ is give by  $a: T(A)\to  A$, $\delta: cT(A, a) \to cT\circ cT(A, a) $ i.e. $\delta: (T(A), \mu_A) \to (TT(A), \mu_{TA})) $ is give by $T\eta_A: T(A)\to TTA$.
Now a coalgebra of $cT$ is a couple $((A, a), a^1)$ where $(A, a)$ is a $T$-algebra and $a'$ is a $T$-algebra morphism $a': (A, a) \to  (T(A), \mu_A)$   such that $a'\circ  \epsilon_{A, a}=1 $ and $T(\eta_A)\circ a'= T(a')\circ a' $.
If $T$ come from the free groups adjunction $\langle F, U, \epsilon , \eta \rangle: Gr \to  Set$  the condition $a'\circ  \epsilon_{A, a}=1 $
 makes me suspect that the only cotriples  $((A, a), a^1)$ are free-triples (i.e. free groups).
But in the work of J. De Vries (Lecture Notes Math N.540 p.654)  in Th 2.3 (p. 659) he show a monadic functor $TTG \to  TopGr \times  Top$ 
(where $TTG$ is the topological transformation groups category, $TopGr$ the topological groups category,  $Top$ topological spaces category) 
where the algebras of the associate monads are the topological actions $\pi: G\times X  \to  X$.
In Th.2.6 he considered the comonads on the algebras category as in mine initial presentation, and prove that the associated coalgebras are topological actions $\pi: G\times X \to  X$ with a continuous cross section $(S, u)$.
So it seems to me that the problem is not trivial, although it come from  very simple considerations, however I have not found anywhere.
We can iterate this process of iteration, what we get? What about (co)triple (co)homology? ecc.ecc.
I ask: Do you know where (if exist) I can find a study of this issue?
If  not, I  hope it can be useful inspiration for those who have shoulders most powerful than mine.
 A: To answer one question (implicitly) raised in the post, the free group functor $F: \mathbf{Set} \to \mathbf{Grp}$ is comonadic, so here the monad-comonad tower (I do not know the standard name for this, if one exists) goes from $\mathbf{Set}$ to $\mathbf{Grp}$ and then straight back to $\mathbf{Set}$. 
Here is a more general theorem of this type: 


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*For $T$ a monad on $\mathbf{Set}$, the free $T$-algebra functor $\mathbf{Set} \to \mathbf{Set}^T$ to the category of $T$-algebras is comonadic provided that the canonical map $T(!): T(0) \to T(1)$ in $\mathbf{Set}^T$ is a regular monomorphism and not an isomorphism. 


Edit: Here are alternate hypotheses that often obtain in practice: provided that $T(1)$ is not initial in $\mathbf{Set}^T$ and the underlying set of the initial object $T(0)$ has at least one element, the algebra map $T(!)$ is a split monomorphism and therefore a regular monomorphism, whence $F: \mathbf{Set} \to \mathbf{Set}^T$ is comonadic. 
This can be extracted as a consequence of proposition 6.13 (page 24) of this paper of Mesablishvili, for example. 
