I'm a big advocate of multiple screens, or screens so huge that the idea of "fullscreen" is pointless. I like a view. Personally, I think that screen real estate is a major limiting factor on how well we "think" with computers - anyone who has done corkboard brainstorming is familiar with how much this sort of thing matters.
But right now I'd like to talk specifically about games on multiple screens. They already exist, of course, with Nintendo being the major purveyor.
In most Nintendo DS games, one screen is where all the action happens, and the other is a kind of status screen. Your stats, or the local map, or other status information. The idea is that you can refer to the other screen at a glance if you need that information. Of course, you usually don't. It's extremely rare that the "information screen" actually needs to be referenced live.
Some games instead make one screen an input screen and one screen an output screen. This is because of the hardware of the DS, which has one touchscreen and one normal screen. Personally, I don't like this very much, either: without tactile interfaces, it's almost impossible to use a touchscreen without looking at it, and the DS touchscreen is often very fickle about exactly where it thinks you clicked.
The upcoming WiiU is vaguely interesting because, like attaching gameboys as controllers, it gives each player their own screen as well as one large shared screen. But, for one player games, this is simply exactly the same situation as the DS: some buttons, a touchscreen, and a non-touchscreen. Size notwithstanding.
Many of the DS games which ended up being most compelling for the double screens were simply those that used the top screen as an extension of the bottom screen, giving you one long super-screen like an old arcade game. This, of course, has nothing to do with the fact that they are two screens - it could just as easily be accomplished by having a single screen that is twice as tall. Anyway, this won't be very possible on the WiiU, because the screens are not in fixed position with each other and not even vaguely the same size.
Still, the advent of multi-screen situations outside of the tiny handheld device market is reasonably important. I've been developing multi-screen toys on my own for a long time, since I have multiple monitors... But that's not the sort of thing most other people have, so it's kind of pointless. It'll be interesting to see whether the WiiU developers end up doing some of the same things as me or not.
One thing I have come to find is that the "details" screen does have an interesting variant. I call it the "context switch" situation.
Let's say you're playing a 2D RPG where you wander around the map as you see fit. Normally, when you approach a person or item, you press the "interact" button and some text box pops up to tell you that you grabbed it or they said something or whatever.
However, if you have two screens, it is perfectly reasonable to have the bottom screen be this navigation view while the top screen is the "what your character sees right now" view. IE, if you walk up to a bookshelf, you don't press "interact". The top screen automatically shows the bookshelf, including the book titles and so on. If you walk up to a person, you see a closeup of them on the top screen, including their facial expressions and body language. (Speech bubbles should ideally appear on both screens, so they don't require a player to switch which screen they're looking at.)
If you like, you can simply swap the screens, switching contexts. Now you have the person or bookshelf on your primary screen and the navigation screen is on the secondary screen. This allows you to do complicated dialog systems, pixel hunts, machine programming, or whatever.
"Wait, that's not fundamentally better than, say, a popup on a single screen!"
A third-person view substitutes for our personal spatial awareness. First-person jumping puzzles are difficult even if the same puzzle would be a snap in real life. That's because in real life, we're aware of our surroundings. In a third-person game, we get a lot of that same kind of awareness out of the game screen. Covering up the game screen effectively blinds that artificial eye.
Minecraft uses popup menus for all its crafting, which is okay-ish because Minecraft is first person: your situational awareness is already crap, you can't get any more blind. But Terraria, with its third-person view, carefully puts its crafting popups at the edges of the screen where they won't interfere much with your artificial situational "eye".
Unbroken situational awareness may not sound like much, but it's actually pretty important. It's the same reason why corkboard brainstorming is often very effective: you have that important detail off in the corner of the board. Even if you aren't looking at it, you can see it in the corner of your eye and you know it is there. It is a piece of the mental terrain. If you walk away from the corkboard and go to lunch, when you get back you'll probably have to spend a few minutes recovering that awareness.
So, some of my toys have been experiments in retaining situational awareness even when contexts change, and I think that's something that most games don't bother with. I'm hoping to see it become more common.