I've always been impressed by the ability of songwriters in "popular" styles - whether that be rock, pop, country, metal or whatever - to write songs which incorporate harmonic and rhythmic elements outside of the usual "predictable" chord progressions and rhythms, whilst remaining accessible and writing the kind of thing non-musical people can sing or whistle along to without difficulty. The Beatles were great at it of course, and Steely Dan comes to mind. Stevie Wonder, Joni Mitchell, another two. I'm not talking about modern jazz, classical, prog rock or any other styles which go out of their way to sound esoteric and unusual. Just catchy tunes which twist and turn and modulate in usual ways but which still remain accessible and catchy. I live for music like this. 
 
Two that immediately spring to mind are from Radiohead, "Knives Out" and "The Pyramid Song." Knives Out uses pretty standard chord types (major, minor, m7, m6 etc) but modulates in interesting ways - it's both dark and disturbing yet immediately catchy, to my ears anyway. The Pyramid Song has very beautiful chords which aren't that unusual, but the rhythm of the song is very unusual whilst still being the kind of thing you can nod your head to. Would be interested to hear of other examples. I'm constantly trying to incorporate usual musical elements into my own music without sounding "nerdy," and it's hard!
 


 
 
 
 

"Turn It On Again", by Genesis.
 
Catchy song, but lots of odd meters.  Apart from the fact that the main riff is in 13/4, there are lots of other odd bars - 5/4, 3/4 etc.
 
What always impressed me was how they made it sound so "straight", mainly thanks to Phil's drumming.
 
I could talk about the devices etc. used in this song for hours, but I won't.

Yes Turn It On Again is a great example, instantly recognizable but very odd meter.

Beatles did a lot of this type of stuff.  Time signature changes happen in a lot of their songs, example:
 
"All You Need Is Love".
The main verse pattern contains a total of 29 beats, split into two 7/4 measures, a single bar of 8/4, followed by a one bar return of 7/4 before repeating the pattern. The chorus, however, maintains a steady 4/4 beat with the exception of the last bar of 6/4 (on the lyric 'love is all you need'). The prominent cello line draws attention to this departure from pop-single normality,

Some of the Beatles stuff has a lot in common with folk music in the way the meter changes to fit the words. You'll hear folk singers adding and removing bars when it suits them, sometimes on a whim, sometimes to fit a lyric that wouldn't otherwise fit. The great singer/guitar player Martin Carthy (from whom Paul Simon stole his arrangement of Scarborough Fair) once maintained that his music was in 1/1 time.

'Solisbury Hill' is another, very sing-along-able, but in 7/4. Likewise Sting's 'I Hung My Head' which is in 9/8. I recall from a documentary that he did have problems getting the Memphis Horns to play in the right places though.

Just because the song is "accessible," doesn't necessarily mean it's legal!
 
HTH. 

Can't all songs just be broken down into series of 1/4, 1/8 or 1/16?
 
Yeah, I'm a bazz player.

'Take Me Out' by Franz Ferdinand does something quite unusual at 0:53, almost like they stitched two different songs together.
 

I dig that stuff. Even something as simplistic as Nirvana's About a Girl which is all Em-G, but then shifts to a C#-F#. Very simple, but it sort of throws you off a bit at first.
 
Rhythmically, I always preferred the opposite - that is, that any odd time should be integrated so that the average listener doesn't even notice. Most people have no idea what 7/4 is, and yet they all follow through Pink Floyd's Money. To me, that's the ultimate goal.
 
As mentioned, the Beatles were masters at that. Led Zeppelin also had a few good ones, like The Crunge. 
 
Sometimes, that carelessness for time signature can be pushed farther - (Have you ever been to) Electric Ladyland. I never even bothered to count that one it just seems to ride in Hendrix's wake...
 
The intro to SRV's Wall of Denial - another one I never bothered to count, could very well be a straight 4 oddly organized, at any rate it's spicier than your average 4/4 - I love playing that riff, it's so groovy.