Over the past several years there have been many MMORPGs of varying success, the most notable being World of Warcraft and the quickly growing Star Wars: The Old Republic. What I find surprising is how the MMO aspect hasn't easily transferred to other gaming genres such as the FPS.
What I imagine is a first person shooter akin to Call of Duty and Battlefield 1942, but set in a persistent world where you can jump in and out of ongoing battles as you go. Now if this could be combined with elements of Minecraft or something similar so you could create your own defences and dig trenches that would be amazing. It is true though that this would then become close to being an RPG, though technically it would be limiting the RPG to PvP only.
The closest any game has yet come to this is when the browser based FPS, Man vs. Machine set the world record for the most players in a single game. This however was a one off attempt. Now what I imagine would be really cool would be a game set during World War 2 where the persistent world is actually the world as it was in the 1940's, with ability to drive military vehicles, drive boats and pilot planes. You would have to be forced to be either Allied or Axis, and have fast-travel to other parts of the world / country, so you could take part in conflicts all over Europe, or even in the Pacific theatre.
I guess part of the issue why such a thing doesn't exist are the mechanics behind player progression, etc. though I think an XP system similar to online multiplayer in CoD and others such as R6:Vegas would still work. Maybe weaponry and ammunition would work the same as RPGs too, and have what you're allowed to carry limited by what class you are. So you maybe a class that "doesn't know" how to fly a plane, or maybe one that does but then doesn't know how to operate a mortar. Maybe this is something where a player could learn skills over time to gradually be able to use more.
Maybe one day there will be massive 1000+ player battles on a regular basis.