There are billions and billions of them.

Monads arising from adjunctons between graphs and sets:

* The monad which takes a graph and turns it into the complete graph on the same vertices.
* The monad which takes a graph and turns it into the discrete graph on the same vertices.

The monad arising from adjunction between graphs and categories (you have to do this right to avoid loops):

* The monad which takes a graph and creates a new one with the same vertices, but connects two vertices iff there is a path between them in the original graph.

Some other random stuff:

* The monad arising from $\pi_0$: it takes a graph and returns the discrete graph whose vertices are the connected components of the original graph.

I have somehow managed to give only monads that are closure operators (the multiplication is an isomorphism). I will let someone else list some other monads.