OP: What the hell is a groove anyway? This is becoming a little elastic, no? OK so blues might be a groove, but we don’t want to list 10,000,000 songs here. But it is!
How about Terry Riley “In C”
::mopping nostalgic tears:: Woulda been a little groovier minus the two false-speeding-up bits.
The beat’s also the same throughout too, but at times it seems (for instance right at 8:08) it might slightly meander into a different style?
A sniggering yea.
Good rock example - it’d be nice to see more of these.
If you really try to dissect the term “groove” there will be off-shoot definitions from that, which is why the parameters stated in the OP should suffice - I’m just trying to home in on one particular aspect of “groove”. Ever since my last post I’ve gotten a few “this video not available” error messages from Youtube when checking out others’ input, so to reign in the elasticity will be a certainly protean exercise here. Sure, the blues (and electronica) definitely lend themselves to basic groove concepts, but I’m pretty confident this thread will peter out before all those possibilities are exhausted. Terry Riley’s “In C” is a good borderline example - where the groove (while lasting the entirety of the song) is a tenuous piano line that’s otherwise swamped by lots of shifting in-and-out sections.
Yeah, it actually rides the groove for only about eight minutes, actually. It goes free-form freakout after that.
Mother of Pearl - Roxy Music There’s the beginning rockin’ part then is slows down to nice steady groove until the end.
ok that was drunk - nm
Daft Punk Is Playing At My House - LCD Sounsystem
Yes We Can - Lee DorseyMiM
Arbouretum - All that will be has become, all that has come isn’t gone
Mysteriously missing from the youtube. 
First hit on google.
“Boogie Chillun”, and its descendant “La Grange”.
Does “Spooky” by Atlanta Rhythm Section fit the spirit of the question? They do a key change/elevation about halfway through, but it still feels like an extended jam.
Well, of course you would find it faster than me! What? Are you using tin-cans instead of Dixie Cups?
I suck so hard sometimes…
Along with the key change/elevation there’s not only one but two guitar solos plus a keyboard solo that definitely chop up what would otherwise more or less qualify as an extended jam, especially how they smoothly work the refrain “Love is kinda crazy…” into the ends of the stanzas to keep it from being a distinguishable chorus.
One groove…
SchoolBoyQ: Collard Greens (NSFW lyrics/video)bobot, bobot
Bo-bo-bot, bo-na-na fanna
Fo-fo-bot. Fee fi mo-mo-bot, bobot!
Tucker!
That’s the nicest thing anyone’s ever said about me! 
Not mentioned, but I think these might qualify:
“Uneasy Rider” by The Charlie Daniels Band
“Genius of Love” by The Tom Tom Club
“Bittersweet Symphony” by The Verve
“No Anchovies Please” by the J Geils Band might also qualify … can’t remember every nook and cranny of the instrumentation. Definitely no chorus or refrain, though.
Out of that bunch, “Genius of Love” might be the only problematic one because of all the different variations in the vocals.
Fucking annoying that “No Anchovies Please” can’t located anywhere on the internet.
Eddie, how about Yello’s “Oh Yeah”? I guess “Oh Yeah” is a refrain, but that song is locked into a pretty small box.
The Doors - Riders On The Storm
Sheesh, it’s like I don’t even know you people!
Mazzy Starr, “Fade Into You”.
Interesting question. Lots of answers, but it takes a bit of thinking.