Last night, Chung-Chieh Shan posted an example of a pointed-set monad on his blog, which happens to be isomorphic to a non-empty stream monad with a different emphasis.
But, I thought I should point out that the pointed set that he posted also has a comonadic structure, which may be exploited since it is just a variation on the "zipper comonad," a structure that is perhaps more correctly called a "pointing comonad."