Songs that don't sound 'radio-friendly' but still get radio play. Changes, Yes - The whole beginning part with the odd time signature. Money, Pink Floyd - The whole song is in 7/4 until the long guitar solo at the end. Spirits in the Material World, The Police - Sounds like an odd time signature (even though it isn't). Barracuda, Heart - Again, it has a strange sounding time signature. They drop a beat every once in a while for no apparent reason. I know there are more that I can't think of.
"Don't Let's Start" - They Might Be Giants Weird chord changes and tempo shifts for such a radio friendly song
Since the Police reuniting makes it topical, 'Roxanne' was a real paradigm buster back when ya'll were in diapers. (and in the same vein even though I never liked it, 'My Sharona') Clapton's 'I shot The Sherriff' had to be the first reggae beat to ever hit the rock airwaves. And since no one listens to the radio anymore, I thought Jet's 'Are you going to be my Girl' was an interesting breakout, coming out of a commercial.
"Heart of Glass" by Blondie repeats this melody in 7/4 a few times before switching back to 4/4. This sounds cool but it's a dance song! Do you think they dropped the beat as a joke on the dancers?
Talking Heads - Once In A Lifetime. Wierd composition, if it you here it on a commercial it's always only certain parts of it.
Satellite is kind of a weird one too and it got radio play when it came out. It switches back and forth between a 3/4 feel and a 6/8 feel. Good one. It's not real clear where the '1' is during the chorus. Tom Sawyer, Rush - Well, pretty much any song by Rush that got radio play was a little odd.
I don't know if this counts as a "radio friendly" song - but it gets played on Rock radio all the time... In Kashmir by Zepplin, the music is in 6 but the drums are in 4. I think that's cool... All You Need Is Love by the Beatles has a 7/4 section. Many Rush songs have some weird time signature in there somewhere. Freewill alternates between 6 and 7 in the verses, Limelight goes back and forth between 7 and 6 and 4 and 3 (depending on which riff they are on!), The Spirit of Radio has a break in 7/4. The cool thing (IMO!) about Rush is that they can do odd time sigs but they still groove. If you don't count them out you don't notice something sounds weird. Unlike Heart of Glass or All You Need Is Love where, to me, the dropped beat really stands out as odd.