Oops I swear I didn’t notice the OP already mentioned A Day in the Life.
I thought it was the name of one person: Brenda Renetti.
The late Richard ‘Dimples’ Fields’ She’s Got Papers on Me.
The first part of it starts off with him in the shower, thinking about his sweetie (who is not his wife). After a couple of minutes of this, his wife (Betty Wright) makes a surprise appearance back in the house because she had forgotten her sweater, and after having overheard his singing, the rest of song is her rant about how she’s getting ready to rid his sorry @ss.
(paraphrasing: ) “Now sit down right her and have a seat. . . but don’t get too comfortable, 'cause you ain’t gonna be here long. 'Cause this is where you used to live.”
A special case, deliberately made as two songs that could be combined into one
For the Fleischer Studios 1939 full-length cartoon motion picture Gulliver’s Travels (their response to Disney’s Snow White and the Seven Dwarfs) they changed the cause of the Lilliput-Blefuscu War from a dispute overwhich end of the egg ought to broken to a debate over which nation’s song should be sung at the wedding of their prince and princes. King Little of Lilput opted for Lilliput’s song (naturally), “Faithful”,
while King Bombo of Blefuscu held out for Blefuscu’s “Forever”
The eventual solution, was to sing the two together as the two-part “Faithful Forever”
The songs were composed by Leo Robin and Ralph Painger, who had a lot of roadway credits. I think the double song idea is a clever one, and not fully appreciated. Gulliver’s Travels was nominated for a “Best Score” Oscar, but lost to “The Wizard of Oz”.
“We Can Work it Out” is another Beatles club sandwich; the John part complements but has a distinctively different tone and attitude than the Paul part.
Laura Nyro was famous for composing songs that ride off in all directions. “Timer” is a good example of five or six different songs shoehorned together into a very satisfying whole.
(“Timer” was the name of Laura’s cat in 1968.)
Are there regressive bands that only manage half a song? 
Maybe that’s punk rock?
Led Zeppelin’s Heartbreaker and Livin Loving Maid are usually played on the radio as one song, but on the album (Led Zeppelin II) they are separate.
Irving Berlin was famous for his contrapuntal songs, where one person would sing a song, another would sing another, and they’d then sing simultaneously.
Well a couple that are truthfully medleys but most people know as one song:
George Thorogood’s “One Bourbon, One Scotch, One Beer” is really a medley with “House Rent Boogie”
The version of “Just a Gigalo” most everyone is familiar with is really a medley with “I Ain’t Got Nobody”.
The first thing that popped in my mind was Canadian one-hit wonders The Kings with “This Beat Goes On/Switchin’ To Glide”
You guys have mentioned many of my favorites, so I will nominate Elton John and “Funeral For a friend/Love Lies Bleeding”.
Man that takes me back to listen to it. This 11 minute song has everything: church organ music, swelling orchestral riffs, hot guitar licks, all tied together with Elton and Bernie's absolute mastery of lyric writing. You don't have to guess at what these lyrics mean, they just tear at the heart.The version of the video is great with photos of band members and performances fading in and out plus the lyrics on top so you don’t miss anything.
Dennis
Just a gigolo - Louis Prima
Question - Moody Blues Oh Well - Fleetwood Mac
Ditto “Band on the Run.” There’s the intro, then the “if we ever get out of here” part, then the main part of the song.
Like you say, it’s something Paul does.
Do songs that switch in the middle, then switch back, count? I’m thinking of Gypsy’s “Dead and Gone”, which has this weird lounge-singer-y interlude in the middle of an otherwise rocking song.
I don’t know if that part is an actual third song. I’d call it the bridge, myself.
Jackson Brown’s The Load Out and * Stay * are usually played together as they were intended to be one song. I like how they sound together, both lyrically and musically.
Any DJ who plays Suite: Judy Blue Eyes without an immediate segue to Marrakech is dj’ing wrong.
Same if they play Hazy Shade of Winter without At the Zoo as a chaser.
Poland Whole/Madam I’m Adam by The Tubes
Switching to a contrasting part in the middle and then switching back is one of the normal structures for a song. But should a song be considered not normal if the sections are so different that they seem to have nothing to do with each other? There’s no absolute rule, it’s a matter for opinion and argument
The most famous version of Benny Goodman’s Sing Sing Sing (With A Swing) from the 1938 Carnegie Hall concert doesn’t credit it, but it’s actually a medley of Louis Prima’s Sing Sing Sing combined with Fletcher Henderson’s Christopher Columbus. Since they both have the same tempo and blend together well, hardly anybody ever notices. Fletcher Henderson was working as an arranger for Goodman at this point, so he probably threw it together himself.