Monads on the category Set of sets and functions are somehow fundamental objects of category theory, and moreover they have important applications to computer science. We know of a good number of monads on Set, but they all appear (at least to me) as isolated examples (other than three big classes of them I'll discuss below). I'm wondering what is known about the category of monads on Set; call it $Mon$$Mnd$. Below, I'll refer to a monad $(M,\eta,\mu)$ in $Mon$$Mnd$ simply by its functor, $M$.
Clearly, the category $Mon$$Mnd$ is not small because for any set $E$, there is a monad $X\mapsto X\amalg E$ and another monad $X\mapsto X^E$. It would be nice to "classify" monads so as to see these as two, rather than as a large number, of examples. Are there ways to "classify" objects in $Mon$$Mnd$ to notice more broad patterns? Here's another class of them for example: those coming from algebraic theories (e.g. the free group monad, the free ring monad, etc.)
For what types of diagrams does $Mon$$Mnd$ have limits or colimits? Clearly $Mon$$Mnd$ has an initial object (the identity monad) and a final object ($X\mapsto\{\star\}$).
Suppose given a monad $M\in Mon$$M\in Mnd$, and suppose we know (1) the set $M(\emptyset),$ (2) the set $M(\{\star\})$, and (3) the function $M(\emptyset\to\{\star\})$. Can we say anything else about $M$? Can we limit the three possible answers to (1),(2),(3) above for monads?
Are any other interesting facts known about $Mon$$Mnd$?