Really fun thread.
I think "Turn it On Again" by Genesis is partially, but not entirely, in an unusual meter, but I didn't see it mentioned yet. I've never seen the sheet music, but I think it's in /8, with bars of 8/8 and bars of 5/8 next to each other, but not consistently, so I don't think you can call it 13/8.
"Solsbury Hill" is a great example because it's so well written that most people (including me for years and years) don't realize that it's in (I think) 7/4. In contrast, most songs in 5/4 immediately sound like they're in an odd meter. Two good examples - although not pop/rock/rap - are the Mission Impossible theme and "Take Five" by the Dave Brubeck Quartet. "Seven Days" by Sting (5/4 or (3+2)/4; I think it's "additive" so the latter is more correct) also sounds like it's in a "strange" meter from the get go.
As a Yes fan, I hear unusual meters fairly often. "Roundabout" and "Heart of the Sunrise" come to mind right away, as does "Siberian Khatru" from _Close to the Edge_, which has some time changes and even a nifty polyrhythmic section with the guitar in a different meter than the bass and drums. If anyone really knows the deeper Yes catalog, "Sound Chaser" from _Relayer_ is at least partially in 5/4, and also sounds "off."
I believe that some of "Tom Sawyer" is in 13/16. Actually, I just listened to it now for the heck of it and I heard some 7/8 in the instrumental section as well. That's one Rush may just play by feel.
"Spirits in the Material World" sounds like it's in an unusual signature, and I've never seen the sheet music, but I think it's just in 4/4 with some really interesting, very deceptive syncopation in the verses.
I saw "Everybody Wants to Rule the World" mentioned. I love that song and have never been able to figure it out; my band played it in grad school and we treated it like it was in 4/4 with a triplet feel; I sure don't think we ever made the tune sound particularly good (much harder vocal than it sounds) but the time signature seemed fine. It may be in 6/4 as written. Anyone seen a transcription?
Finally, two examples from TV themes: the theme song to ER was in 3/4, and there's either an irregular couple of bars or an odd meter to the Amazing Race theme that I've never doped out correctly. Oh, and I think that the main theme from The Terminator is in some cool/bizarre signature, but it's hard to tell from the movies unless you have the score and can hear it unedited, which I don't and haven't.