There are billions and billions of them. But it turns out I originally suggested two non-examples:

* The non-monad which takes a graph and turns it into the complete graph on the same vertices.
* The **comonad** which takes a graph and turns it into the discrete graph on the same vertices. (This example was edited after Andreas Blass made his comment.)

And two that still seem to be examples:

* 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.
* 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.