I'm interested in saying something about the colimit $$ T_F(X):=\varinjlim\Big(X\to FX\to FFX\to\dots\Big) $$ taken starting from a pointed endofunctor $F\colon\cal C\to C$; fiddling a little with adjointness, this amounts to a colimit $\mathbb N\to [\cal C,C]$, which exists as soon as $\cal C$ is cocomplete.
It seems to me that the functor $T_F\colon \cal C\to C$ sending $A$ into $\varinjlim{}_n \;F^nA$ is again pointed, and it admits a canonical arrow $T_F\to T_FT_F$ (deduced from UP of colimit and the arrow $\varinjlim \circ F\to F\circ \varinjlim$). I would have expected either a copointed functor with something to be proved to be a comultiplication, or a pointed functor with a would-be multiplication. But this?
I do believe this problem is elementary, but in general one can hardly say anything. Nevertheless I'd like to see something explicit: feel free to give me additional (non-too-much-trivial) assumptions.
A lazy computation: $$\begin{align*}T_FT_F(X) &= \Big(\varinjlim_m F^m\Big)\Big(\varinjlim_n F^n\Big)X\\&\cong \varinjlim_m F^m\Big( \varinjlim_n (F^n X)\Big)\\&\cong \varinjlim_m\Big(F^m \varinjlim_n (F^n X)\Big) \\& \leftarrow \varinjlim_m\varinjlim_nF^mF^nX\\&\cong \varinjlim_{(m,n)} F^{m+n}X \cong \varinjlim_{(n,n)} F^{n+n}X\\&\cong \varinjlim_n F^{2n}X \cong T_FX\end{align*}$$ where the last isomorphisms hold since $\omega$ is a filtered category, hence sifted, hence a series of finality arguments help to conclude.
I suspect that your lazy computation is a little too lazy. Try writing down the actual maps in these colimiting diagrams, and I think you'll see that you need $F$ to be well pointed.
In that case, if you assume additionally that $F$ preserves sequential colimits, so that your map is an isomorphism, then $T_F$ is the free monad generated by $F$, which is idempotent (i.e. its multiplication transformation is an isomorphism) since $F$ is well-pointed.