TITLE: Non-monadic adjunction
QUESTION [6 upvotes]: Could someone give some examples of a non-monadic adjunctions please? Possibly explaining why they are not monadic and how they contradict the monadicity theorem?
Thanks!

REPLY [3 votes]: This might not be the answer you are looking for, but it could give you a little insight on what monadic really means.
Given a monad $(T,\eta,\mu)$ on a category $\mathcal C$, you can form the category $\operatorname{Adj}(\mathcal C,T)$ of adjunctions above $T$:

its objects are the ajdunction $F: \mathcal C \rightleftarrows \mathcal D :U$ ($F$ on the left) such that the induced monad $(UF,{\rm unit},U\,{\rm counit}\,F)$ equals $(T,\eta,\mu)$ ;
a morphism from $F: \mathcal C \rightleftarrows \mathcal D :U$ to $F': \mathcal C \rightleftarrows \mathcal D' :U'$ is a functor $K \colon \mathcal D \to \mathcal D'$ such that $KF = F'$, $U'K = U$ and $K$ commutes with counits.

This category $\operatorname{Adj}(\mathcal C,T)$ has a terminal object: the Eilenberg-Moore category of $T$. The unique morphism from any other object is the so called comparison functor. Hence, by definition, a adjunction $F: \mathcal C \rightleftarrows \mathcal D :U$ is monadic if and only if it is terminal in $\operatorname{Adj}(\mathcal C,UF)$.
It should make you realize that, in a sense, most adjunctions are non-monadic. For example, the previous category $\operatorname{Adj}(\mathcal C,T)$ always have an initial object: the Kleisli category of $T$. And this category is terminal only if every algebra over $T$ is free! Which is quite uncommon. It produces a lot of examples of non-monadic adjunctions $$Set \rightleftarrows \text{Free algrebas over T}$$
for $T$ the monad of groups, monoids, modules over a non-field ring, etc.

Note that the initial and terminal object can be the same (as in the example of Captain Lama), but still leaving some "room" for non monadic adjunctions in between.