What foods *require* a food processor?
For starters, you'll almost certainly need a food processor for anything you need to mix dry, unless you've also got a heavy duty coffee grinder that you don't mind cleaning inside-out thoroughly and frequently.  Conventional stationary blenders, at least in my experience, are terrible at this task, and immersion blenders aren't designed to be run dry (in fact, most of the good models come with stern warnings never to do this).

Examples of this are: Nut butters, fresh herb purées, meat purées, and homemade bread crumbs and flours.  Pesto sauce also sounds like a long shot, but maybe there are customized recipes.

I've also never had any luck making cheese sauces, cheese dips, or other cheese preparations in a blender, and haven't had much better luck with my admittedly cheap stick blender.  It's difficult enough with a food processor, but usually with them it's just a matter of scraping down the sides a few times.  Whenever I've tried to make a manicotti filling, a mayonnaise-based tonnato sauce, a blue cheese dip or a goat cheese mousse with a blender, I've been totally unable to get a vortex going and have had to resort to the food processor to thin it out (after which I might transfer it back to the blender, but I usually don't bother).

It's an issue of viscosity, I think.  Too viscous, and conventional/stick blenders are practically useless.

Now a lot of this might be possible with very high-end equipment.  A Vita-Mix or Blendtec can blend just about anything, and usually comes with a tamper so you can force dry or viscous food down into the blades without risking life and limb.  Also, some very high-end immersion blenders - and when I say "high end", I mean that they are labeled as "power mixers" rather than "immersion blenders" - can be run continuously and possibly even dry.  But I would not want to try this on a consumer-grade stick.

I've also heard that you can do some of these things with a Magic Bullet.  For those not familiar with the product or infomercials, it's basically a really tiny blender that you can pick up and shake around while it's running - ergo, much easier to do dry preparations.  But the effort required is still much higher than that of a food processor.

I think that pretty much covers it.  If you need to purée anything solid or even semi-solid, you'll likely regret not having a food processor.