(+) is assuredly a monoid. But so is (*)! (Try looking at it through logarithms.) Unfortunately, a given type can have only a single instance of a given typeclass.
There is in fact a way to design Num "rationally", via number theory; I suggest you at least familiarize yourself with that. (This will bring you face to face with the above issue, as it turns out that Num requires two monoids.) It turns out that this is not necessarily a good idea. But see http://hackage.haskell.org/package/numeric-prelude for an implementation of it.
Well, at some point you will have to make concessions. I for one, would love to use (+) without Num or going through an entire semester of Algebra. And DreamApart's proposal seams a reasonable compromise. I don't see it being included into base any time soon however, as it would break an enormous amount of code. As for the Maybe instances. From afar, if you think of Nothing as 0 and Just x as (not 0), the Monoid instance looks like (*) and the MonadPlus instance looks like (+). Also, I would prefer. class (Monoid m) => Ring m where ... class (Group a, Ring a) => Num a where ... Cheers Silvio