I see that Can we have some sort of notification system? answers "NO" in no uncertain terms - I'm going to take a shot here because what I'm suggesting should not be any more complex than the logic it takes to maintain a browser session that makes the little number of messages appear in the StackExchange dropdown...
I think this could also be done with a scripted browser control, but it would be cleaner if one could simple say GET /user/notifications or something?
If you have a valid session cookie for a given user, you actually do have access to the messages in the Stack Exchange dropdown using the following URL:
http://stackexchange.com/genuwine
This URL returns the user's most recent notifications across the network as JSON data.
Keep in mind that this would only be useful in a situation where you have access to a user's session cookie - such as in a browser extension or addon. Apparently there is reason to hope that this will eventually be implemented when the next version of the API rolls out.
