Is there a way to override notifications that don't let me dismiss them? I have a Galaxy Nexus that's running 4.2.2. Certain apps, like Words with Friends, occasionally send spammy notifications for trying out a different game or something like that. They won't dismiss when I press the dismiss all button, and I can't manually swipe to remove them. I have to actually press the notification which takes me to the app.

Obviously I don't want to have to bother opening and immediately closing an app to remove a notification. The simplest solution of course would be to not install apps that do this, but in general, is there a way to have more control over your notification bar?
AFAIK, with Android 4.2+ you can regulate on a per-app-base whether you want to see notifications or not. Long-pressing a notification should open a menu, from which you can navigate to the corresponding place, and define you don't want to see notifications from that app anymore. (As I have no device with 4.2+, I cannot check myself).

On the other hand, "spammy messages" sound like they come from some advertizement module, such as Airpush or Leadbolt. You can verify that with Ad-Scanners (I recommend AppBrain Ad Detector). If I guessed right with that, you have two ways to react on this: for Airpush, there's a way to Opt - out, same seems to apply for Leadbolt (Opt-Out). AppBrain Ad Detector clearly shows you which network is involved, see below screenshots.

 
AppBrain Ad Detector (source: Google Play; click images for larger variant)

Or you consider avoiding apps using such an aggressive advertizement, uninstall yours, and watch out for a less offensive alternative -- that's what would be my choice. I have full understanding for devs to fund their fremium apps using ads -- but please, leave those adds where they belong: inside the app they are funding. Pushing ads to the notification bar, or adding "funny shortcuts" to the user's homescreen, is intruding the user's privacy, and has no connection to the app at all (in most cases, users even have no clue where that "stuff" comes from).