One possibility would be improving the color range, even if the resolution isn't improved. Rather than cramming in three phosophors per pixel, perhaps we could have four, or more. There's a considerable chunk of color space not well represented by RGB color.
I don't know how much of a difference it would make to TV viewers or gamers, but I know that artists would be grateful for a better color range. The conversion from RGB to CMYK is always a bit of a crapshoot; things that look great on your screen don't look as good when they come back from the printers, and there's a whole range of stuff it doesn't occur to you to try because you can't see it.
I could even imagine that it might be handy for medical imaging and other applications where you want to cram as much information onto the screen as possible: more pixels may not improve things but more colors might. Though more pixels could achieve that as well: it would be nice to be able to zoom in by bringing your face closer to the screen without simply seeing bigger pixels. Head motion is kinaesthetically appealing: you can move in and out without losing your sense of overall place.
Sharp already makes a four-pixel TV, with an added yellow (which is especially helpful in skin tones). I think it would be neat to be able to produce true indigo, violent, and cyan. If this lets you add more phosphors without costing resolution, it might not be a killer app, but it could be a desirable thing.