A very Cinematic Devil May Cry Style game with a little bit of Character indication that something is wrong but the Hud moved to the controller making for a very Cinimatic presentation
If I could play WoW without the Hud I would but I'm not that good.
Super Duper Scribblenauts.
A versus Puzzle game where each person has a puzzle on their respective screen that manipulates their vs entity on the main screen.
All the GBA Connection Games.
A platformer that you played on the big screen but the little screen would have bubbles you had to pop, or something similar, before it leaves your little screen and effects the big screen characters. Later on some of those will be good and needed, making it hectic.
A Good RTS,
Though for Kicks and giggles StarCraft2Elite Beat Agents Super Awesome Extended Cafe Edition
Simon Says,
Dut. Dut Dot. Dut Dot Dut. Dut Dot Dut Dop. Dut Dot Dut Dop Dop.Mario Party Mini Game Vs.
Pokemon Stadium. Literally, Pokemon Stadium.
<Put in generic play based Sports game here.>
Gauntlet.
Warioware minigame game.
You know if it has motion sensing and knows where the TV is than a game where you can bring the screen up like your looking through it and it acts as a detector of sort, like an Xray or MMR.
A Dora game that is an exact reproduction of the episodes because, Dora is really a TV Show about a PC Game. So the three items show up on the big and little screen. Where you need to point. You could do the voice interaction through a microphone. It be a solid game for kids.
Elite Beat Agents.. Oh wait.
Super Mario Time Trials. This would be a versus games that takes levels from Super Mario Bros. 1, 2, 3, Land and pits players in speed runs agains each other. The big Screen would be used to do an announcer style cut to each of the competitors as its going and verbally mentioning how well the players are doing. Also have stats etc. The players themselves would be playing through the level on their own screen. At the end of a round the winner would be declared on the screens and the big screen would show "ghosts" of each player running through the level.
And RPG with Mini-game style combat, think Mario RPG, done on the touch screen for special attacks etc. Be turn base but allow for people to jump in and take extra characters to move battles faster. So with 4 players everyone would control and do the minigames for a character. With 2 then each person be responsible for 2 characters. Then with one obviously they be doing each of the characters each turn and their mini-games.
A racing game with a Hot Potato type Mechanic. So you be racing like F-Zero with friends but you need too keep this ball that shows up on your screen in play between each other or it will weigh down maybe damage your racer.
A gaming feature a Goo based main character which you can shape on your controller to get through different obstacles as you go on the fly.
Angry Bird Vs. It be Angry Birds with a versus mode where each player get a puzzle on their screen.

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I'm going to stop there.
LOL, I thought you were asking for concepts well. You got them anyway.