It’s not exactly a count off, but unusual counting in a song anyway, from Bob Dylan’s “I Shall Be Free No. 10”:
I was shadow-boxing earlier in the day
I figured I was ready for Cassius Clay
I said “Fee, fie, fo, fum, Cassius Clay, here I come
26, 27, 28, 29, I’m gonna make your face look just like mine
Five, four, three, two, one, Cassius Clay you’d better run
99, 100, 101, 102, your ma won’t even recognize you
14, 15, 16, 17, 18, 19, gonna knock him clean right out of his spleen"
The opening to All That Jazz has a nifty count-in.
And it’s like that on the soundtrack album for the nit-pickers who say, “Movies don’t count.”
Duh. Sorry, @CookingWithGas . You said that – about three numbers coming from three different band members. I need coffee 
Pretenders , Middle of the Road, being a countoff in the middle break of the song. I don’t know if it’s true, but I heard they were placeholders for lyrics that never got added in the final version
From far out in left field: Comedian Kathleen Madigan’s father was a lawyer (first 25 seconds).
Cream: A Mother’s Lament: “Are we lovely? A-one-ah, a-two-ah, a-free-ah, a four”
The Police live versions of “Born in the 50s” start out with a similar six count:
The reason, I assume, is that there is a two-beat pick-up bar before the first measure (so the song starts on the 3 instead of the 1) and your options to count off are either one-two and go into the song or count to six and go into the song. With latter you have a little more of establishing the tempo.
Todd has a little trouble counting in…
(He also has a little trouble holding a note, around 2:15 
)
James Cotton “Georgia Swing”
mildly interesting botched count-in by (I think) Todd Rundgren
I was drunker than shit.
“He wanted a 13, but they gave him 31…”
The Offspring were far ahead of their time.
Rock around the Clock - used as the theme song for Happy Days starts with:
One, two, three o’clock, four o’clock, rock
Five, six, seven o’clock, eight o’clock, rock
Nine, ten, eleven o’clock, twelve o’clock, rock
We’re gonna rock around the clock tonight
Then the theme song from a spin-off of Happy Days, Laverne and Shirley starts with:
One, two, three, four, five, six, seven, eight
Schlemiel! Schlimazel! Hasenpfeffer Incorporated
During the first verse of Space Oddity by David Bowie there’s a 10, 9, 8… rocket launch countdown in the background. As it reaches ‘Take Off’ a much fuller instrumentation begins playing.
TCMF-2L
Aw, Warsaw was already taken :(.
I like the rhythm of the count-in for The Police’s Invisible Sun
And Mo Tucker counting in on After Hours is awesome.
Actually, you do have more options than that – you can count is 3-4-1-2 and then the song starts on 3. I have a feeling I’ve heard that somewhere.
There’s a Beatles song (not sure which one) where John starts by mumbling “Sugarplum fairy, Sugarplum fairy” before the band comes in. It’s on the Anthology version but you don’t hear it on the radio. Does that count?
It was this one. Sorry, I’m on mobile and can’t format properly. Beatles’ ‘A Day in the Life’: 10 Things You Didn’t Know – Rolling Stone
Wow, thanks; I haven’t heard that in ages.