Hey, two out of three’s not bad.
Two from The WHo…
This song is over…
I’m singing this note 'cause it fits in well with the chords I’m playing…
Neil Young’s “This Note’s For You.”
Jim Croce’s “I’ll Have to Say I Love You In a Song”
Steely Dan’s “Deacon Blues” (“I cried when I wrote this song. Sue me, if I play too long.”)
Simon & Garfunkel’s “Leaves That Are Green” (“I was 21 years when I wrote this song. I’m 22 now, but I won’t be for long.”)
Joni Mitchell - “Songs to Aging Children Come”
Songs to aging children come…This is one
John Hiatt - “Memphis in the Meantime”
I don’t think Ronnie Milsap’s ever gonna record this song
and the ever popular children’s travelling song:
I know a song that really gets upon your nerves,
gets upon your nerves,
gets upon your nerves.
I know a song that really gets upon your nerves,
and this is how it goes…
(repeat infinitely or until Dad drives the car into a concrete embankment)
Cher - Different Kind of Love Song: “This is a different kind of love song / Dedicated to everyone”
From the soundtrack of the movie “Trick,” Como te Gusta Me Pinga? (pardon my possibly atrocious spelling): “I told my friend the writer / How happy I would be / If he’d write an opening number / Especially for me / But when he had it finished / It came as quite a shock / He handed me a song titled ‘How Do You Like My…’”
The Beatles Only A Northern Song
Prince, 1999:
I was dreamin’ when I wrote this, forgive me if it goes astray…
I love that one.
The Allman Brothers did a tune called Just Another Love Song on their album Win, Lose or Draw. The chorus is as follows.
“Just another love song I’m singing,
And you know people sing them all of the time,
Just another lonesome guitar ringing,
The only difference is this one is mine.”
And Nobody Knows from Shades of Two Worlds is a little self-referential, too. One verse is:
“Poets they come and the poets they go,
politicians and preachers they all claim to know.
The words that are written and the melody’s played,
as the years turn their pages, they all start to fade.”
I just remembered the (in)famous Lambchop song:
And now I’m doing my damndest to forget it again.
DAMMIT, Interrobang!?! You’ll be the doom of us all. There’s also “I am Henry VIII I am.” (And my “I love that one” comment was in reference to Only a Northern Song.)
Randy Newman’s “Rednecks,” from the album Good Old Boys: “So I went to the park and I took some paper along, and that’s where I made this song.”
Pixies “Tony’s Theme”: “This is a song about a superhero named Tony… It’s called Tony’s Theme”
Oh, and another from the Pixies, “Subbacultcha”: “This is a song about something there/there’s just something about this song”
Steely Dan, “Deacon Blues”: “I cried when I wrote this song, sue me if I play too long…”
Three Dog Night’s “Old Fashioned Love Song” is entirely about itself: “Just an old fashioned love song/One I’m sure they wrote for you and me/Just an old fashioned love song/Coming down in three part harmony…”
I’ve got to stop; I’m coming up with examples faster than I can type.
Don’t know the title, just these lyrics: “Just an old-fashioned love song…” (It’s an oldie)
One Down by Ben Folds
“And Andrew came along y’all
To write a couple of lines or so
This is one I finished yesterday
And I’ve got 3.6 to go”
Barry Manilow’s “This One’s For You” (peak Billboard position # 29 in 1976):
Now I have Eric Idle in my head: “I bet you they won’t play this song on the radio…” Cue the silly sound effects.
But that makes me think of Billy Joel’s “Baby Grand,” sung on the album with Ray Charles: “They say that no one’s gonna play this on the radio / They say the melancholy blues are dead and gone.”
In a slightly different vein, in his song “The Entertainer,” Billy Joel complains about what the record company did to, IIRC, “Piano Man”…
“It was a beautiful song, but it ran too long / If you’re gonna have a hit you gotta make it fit / So they cut it down to three oh five.”
Pink Floyd’s Jugband Blues:
And I’m wondering who could be writing this song…
Colosseum’s Jumping Off the Sun, in which the lyrics describe the contour of the melody:
This song goes up and up, down and down and up and down again…
Or, in a similar vein, there’s Matching Mole’s Signed Curtain:
*This is the first verse
This is the first verse [etc.]
And this is the chorus
Or perhaps it’s a bridge
Or just another part of the song that I’m singing
And this is the second verse…*
I Predict by Sparks.
I hate to be the one to bring it up, but it fits the OP.
(Theme from) The Monkees,on The Monkees album (1966), by The Monkees
Got another Oldie (yes, I am on in years myself. That’s why I remember all this old stuff.) 
Creeque Alley by The Mamas and The Papas
The song tells the history of how they got together. It also makes reference to The Lovin’ Spoonful, The Byrds, and Barry Maguire.
Song About Nuthin’ by Powerman 5000:
This could have been about so many things but it’s not
So don’t waste time and look for the meaning
You heard so many things you couldn’t recall
But this time I swear it’s not about anything at all