Brings to mind Tina Turner patiently explaining how it’s going to play out: “we’re gonna take the beginning of this song and do it easy, then we’re gonna do the finish rough. This is the way we do ‘Proud Mary’. And we’re rolling, rolling, rolling on the river. Listen to the story: ‘Left a good job in the city…’”
Bowling for Soup’s Punk Rock 101 has the line “Same song, different chorus” as a bridge(?) between the first verse and the chorus, and “same song, second chorus” between the second verse and the chorus.
Jack Johnson’s Sittin, Waitin, Wishin:
Now maybe you’ve been through this before
But it’s my first time, so please ignore
The next few lines 'cause they’re directed at you.
Two examples from Showbiz Kids by Steely Dan.
They got the heavenly bodies,
They got the Steely Dan t-shirts
Showbusiness kids making movies
Of themselves, you know they
Don’t give a fuck about anybody else
Foo Fighters - The Last Song
After the Lovin’ - Engelbert Humperdink:
*So I sing you to sleep
After the lovin’
With a song I just wrote yesterday
And I hope you can hear
What the words and the music have to say. *
Romeo and Juliet - Dire Straits
All I do is kiss you
Through the bars of a rhyme
*Borrowed Tune *- Neil Young:
I’m singing this borrowed tune
I took from the Rolling Stones,
Alone in this empty room
Too wasted to write my own.
“An Old-Fashioned Love Song” spells out – in three-part harmony – that it’s just an old-fashioned love song, comin’ down in three-part harmony.
Nobody’s mentioned Herman’s Hermits’ “Henry VIII” yet?
Second verse, same as the first!
I also thought of Sly and the Family Stone’s “Dance to the Music” and Archie Bell and the Drells’ “Tighten Up,” which are both songs about playing the song, essentially:
All we need is a drummer
For people who only need a beat, yeah
and
First tighten up on the drums
Come on now, drummer
I want you to tighten it up for me now
The Leonard Cohen “the fourth, the fifth” thing wins by a long chalk. Soooooo clever.
A close second, IMO, is Natasha Bedingfield.
I really like the following lines:
Read some Byron, Shelley and Keats,
Recited it over some hip hop beats.
I’m having trouble saying what I mean
With dead poets and a drum machine.
The intro to Tighten up (“Hi, I’m Archie Bell and the Drells”) was so memorable that it was parodied years later on the Simpsons: “Hi, I’m Archie Bell, and I’m also the Drells”.
This reminds me of Paul McCartney when he was on Unplugged. He started performing “We Can Work It Out,” and he started singing the wrong verse. So he stopped the band, then addressed the audience by saying, “this is so informal, we’ll start again.”
This was put on the “Best of Unplugged” CD that came out several years ago.
I am music and I write the songs…
Not really an actual song, but I think Detritus’s troll jody from Terry Pratchett’s “Night Watch” counts:
Now we sing dis stupid song!
Sing it as we run along!
Why we sing dis we don’t know!
We can’t make der words rhyme prop’ly!
In “Tubular Bells” (probably best know as the theme to The Exorcist, Mike Oldfield announces each instrument as it is played (including, IIRC, a couple of different types of guitars), concluding, of course, with “tubular bells”.
Robbie Williams, Strong.
‘- When I get drunk I dance like my dad,
and that’s a good line to take it to the bridge’
*“This is the song that never ends,
It just goes on and on my friends.
Some people started singing it,
Not knowing what it was,
And they’ll continue singing it forever just because
This is the song that never ends…”
*
I think CCR does that in “Jambolya”
For self-referential songs, the first thing that popped into my head was the first line of John Cougar’s first hit:
“A little ditty about Jack and Diane”
But IMO the all-time cleverest is from a song that’s pure fluff: “Sunshine Lollipops,” by Leslie Gore.
The refrain goes:
“My life is sunshine, lollipops, and rainbows
That’s how this refrain goes
So come on, join in!”
Newish Australian band, called Operator Please, had a reasonably succesful (In Australia) single off their debut album, called “It’s a song about Ping Pong”
Looking it up on Wiki I see “The song was featured in the 16th episode of season 4 of the television series CSI: NY.” 
Two nitpicks: this was first mentioned by Mutantmoose in post #84, who correctly identified the person as Viv Stanshall from the Bonzo Dog Doo Dah Band.