I play old games a lot and even though it's not realistic I wish consoles were backwards compatible with all their predecessor's games. I don't like to re-buy games unless they are enhanced. Halo Anniversary Edition was such an improvement that I was happy to re-buy it, but I don't want to re-buy digital copy of a game that I already own just to play it on a new console. Stuff like the virtual console is great for people that don't have any of those games, but I have to keep my old systems in working order to play my games.
For me, backwards compatibility is very important. The lack of it isn't a deal breaker, but if a system has it I'll use it.
I'm hoping that not only will the Nextbox have full backwards compatibility, much like PCs do, but like PC games the Nextbox will be able to enhanced 360 games with better frame rates and higher resolutions.
At the very least I expect the entire XBLA and XBLIG libraries to be fully playable and transferable on day of Nextbox release.