Spinning Wheel, by Blood, Sweat and Tears. At the end, they end it with a redition of 'Ach du Lieber, Augustine"; when the instruments sort of peter out in a confused mess, the drummer comments “That wasn’t too good”, and you hear laughter from the rest of the band.
Temptation Inside Your Heart– Velvet Underground
Drive-By Trucker’s Mike Cooley does some chatter at the end of *“Daddy’s Cup”, *talking to someone in the studio. He and Patterson Hood do some chatter at the end of another song, but I can’t remember the name! “I’m gonna reel you in!”
I do this goofy thing where anytime this song comes on, I always turn it up! Even if just a little.
This pretty much it.
Oh and I could be wrong for it might be scripted, but Marvin Gaye on “What’s Going On?” and “Got To Give It Up”. Both intros features crowds having a good time conversing.
Mighty Might Bosstones on Desensitized
“You gotta ease up, there, ease up. I can’t fucking hear anything.”
“What the FUCK!”
“Turn your reels down”
Right at the beginning. Inspired.
Cowboy Mouth at the beginning of 'Irish Boy"
“Fred knows what I’m talking about”
Intro and outro of Jethro Tull “Baker St Muse”. The beginning “Shit shit shit, take 2” sounds genuine. The end getting trapped in the locked studio sounds like it is more deliberately scripted though.
The beginning of Spoon’s “Don’t You Evah” includes Britt Daniel asking if Jim Eno has recorded the talkback mic.
XTC - Towers of London. I reckon they did this sort of thing a lot. 
The Great Leon Russell, I Put A Spell On You
Does “Ok” and some guitar tuning count? Yes: And You and I
Reverend Horton Heat’s “Big Little Baby” has a false start with a belch and someone cracking up hysterically.
I think there are probably a few Jimi Hendrix songs with studio chatter (which one is it where he says something about his ear goggles?), but a good example is Voodoo Chile.
The Ocean, Led Zeppelin. Before the song starts, you can hear John Bonham: “We’ve done four already but now we’re steady, and then they went 1, 2, 3, 4!” (He’s referring to four previous, unsatisfactory takes on the song.)
Don’t Bring Me Down, Electric Light Orchestra. Not exactly studio chatter, but at the very end of the song, there’s a “ka-chunk” noise; it’s the sound of a fire door closing in the Musicland studio in Munich, where the album was recorded. That sound was almost undoubtedly an intentional inclusion, to close the song (and the album).
A couple of albums that begin with studio chatter:
The Kinks: **Something Else **(“This is the master, Clive!” “Nice and smooth.”)
Procol Harum: **Exotic Birds and Fruit **(“Is it on, Tommy?”)
IIRC, they’re discussing various cameras.
“Back at the Funny Farm”, the leadoff track from Motörhead’s Another Perfect Day, begins with Lemmy saying: “Uh, a bit, a little, listen…” [random guitar noodling] “Top notch!” In fact, a bunch of songs from that album contain random studio chatter, though it mostly amounts to belches & stuff.
Metallica’s “Blitzkrieg” ends with the band laughing and Lars Ulrich saying: “I fucked up, I fucked up in one part.”
“Rock of Ages” by Def Leppard begins with producer Robert “Mutt” Lange counting down with: “Gunter, glieben, glauten, globen.” Lange would normally give a normal “1-2-3-4” count, but after countless takes he started speaking random gibberish instead. 
Bowie’s song Andy Warhol off the album Hunky Dory has some studio chatter in the beginning when Bowie correct’s producer, Ken Scott’s pronunciation of “Warhol”.
The Replacements’ song “Treatment Bound” features the band members talking to each other about how they screwed up playing the song.
The whole fourth side of Something/Anything is riddled with Todd talking.
For an 80s new wave throwback: Tones on Tail’s song “Go” ends with someone saying, “Let’s do the ya-ya’s”.
Franz Ferdinand’s song “Eleanor Put Your Boots On” has some chatter in the beginning, but I can’t make out what’s being said.
Appropriate. That song makes me want to throw myself into a well.