It is well known that a monad $(T, \mu, \eta)$ can be factorized in multiple ways as adjunctions, and that in some sense, Kleisli is the initial factorization while Eilenberg-Moore is the final factorization. What are examples of factorizations in between?
Identity monad on $\mathrm{Set}$ factors in many different ways. Kleisli and Eilenberg-Moore categories coincide, but another factorization is given by the free functor to $\mathrm{Top}$ (equipping a set with the discrete topology). This example illustrates that one needs to be careful with the intuition that a generic factorization lies between the Kleisli and Eilenberg-Moore adjunctions. Kleisli category always embeds, but the comparison functor to Eilenberg-Moore category is in general only faithful.