I think that the true power of the DS is really only going to be exploited by RPGs. I am just having a hard time thinking of any other game type that would really benifit from a second persistent screen. Anyone?
I actually have a few ideas for the use of a second screen. I've got the good ones written down somewhere.
For instance, imagine a Zelda game like OoT, two different time periods. Now imagine if on one screen you have past and on the other you have present. That alone opens up ALL kinds of possibilities for puzzles, move or destroy something in the past and watch it take effect in the present. And even more interesting you could make the puzzles naturally occuring, doing something to an object that would cause erosion or something(make the player think.) This could be extended to boss battles and all kinds of stuff.
You could even go further with that, and instead of different time frames you have different dimensions. Then you could have a more "real-time" deal where you can directly affect one dimension, instead of an indirect change as in the lapse of time. Also, both dimensions could affect the other adding a whole new level of gameplay.
I won't go into further detail. It'd take too long and I don't feel like collating my thoughts
That is just the tip of the iceberg, though.
The only limit to what you can do is your imagination!