Q. Are We not Men? A. We Are Devo - listened to it 188 times
Abbey Road - 165 times
The Cars - 126 times
EDIT: correction, I did not listen to the song ‘Jocko Homo’ in its entirety on March 3, 1981. So, perhaps in the interests of honesty and accuracy I should count only 187 complete listenings.
Do singles compilations count? Because I’ve listened to** The Cure**‘s Standing on a Beach several thousands of times…and then other compilations in descending order (Louder Than Bombs by The Smiths, Bauhaus 1979–1983, **Jethro Tull’**s Living In The Past (the SA-only compilation not the 1972 one))
If not, then it’s a contest between *Century Flower *by Shelleyan Orphan, Rattlesnakes by Lloyd Cole and the Commotions, Aion by Dead Can Dance and Reading, Writing and Arithmetic by The Sundays, I’ve listened to all those more than a thousand times (like, around once a week for the last 20 years plus, with occasional play-on-repeat binges)
Back in the day, when all I had was a turntable, and you really had to listen all the way through, there were:
Led Zeppelin IV
Dark Side of the Moon
Agents of Fortune
Now that we live in a digital music era, I rarely listen all the way through. Now that I can easily skip Debbie Denise and other such filler songs, I’ll never listen to AoF all the way through again. (I don’t even bother to import the crappy songs, so there is no Within You Without You interrupting my perfect Sgt Pepper album. 
)
Actually, now that I think about it some more “Best of the Smiths” Vol 1 and 2 might also be the winner for me, although it’s only one of the discs about half the time, so it’s hard for me to pinpoint how many times I’ve listened through the whole thing.
Appetite for Destruction, by virtue of simply having owned it the longest; in fact I didn’t own any other album for at least another 2 years. It’s pretty good, too.
“Who’s Next” held the record (no pun intended) to the point where I can do without hearing “Behind Blue Eyes” ever again.
Until I got “Quadrophenia”, which holds the record to this day. There was no better way to try to “cope” with teen angst than with an entire album about teen angst !
Probably something like:
KISS Alive
Goodbye Yellow Brick Road
Made in the Shade
Quadrophenia
The Cars
Abbey Road
Ziggy Stardust
Who keeps track?
Nice choice!
Mine would probably be Led Zeppelin I or II. Man, if I haven’t heard those things a thousand times by now I would be very surprised.
For me it would probably be: Paul Simon - Graceland
Although a couple more deserve honorable mention:
Stray Cats – Rant N’ Rave With The Stray Cats
Bachman-Turner Overdrive – Best Of B.T.O. (So Far)
I listened to this one countless times because it was the ONLY 8 track tape on the tractor I drove one summer. It was either that or listen to the AM radio in the late 70s. The Farm Report just isn’t that interesting.
My most played has got to be the album Eagles “Their Greatest Hits (1971–1975)”.
I looked the album up to see just how popular it was. Looks like it is the number 2 all time seller with 32 million “certified” copies sold.
**Michael Jackson Thriller 1982
46.0
Eagles Their Greatest Hits (1971–1975) 1976
32.2
Fleetwood Mac Rumours 1977
27.9
Whitney Houston / Various artists The Bodyguard 1992
27.4
AC/DC Back in Black 1980
26.1
Pink Floyd The Dark Side of the Moon 1973
24.2 **
Meat Loaf Bat Out of Hell 1977
21.5
Bee Gees / Various artists Saturday Night Fever 1977
20.6
What, no love for A Farewell to Kings? 
That’s my favorite Rush album – probably heard it all the way through forty times, but my favorite three songs from it over a hundred times each (not counting the 20 or so times I practiced “Xanadu” with my high school garage band).
Boards of Canada, Music has the Right to Children, by virtue of more than a decade of listening to it several times per month as I drift peacefully off to sleep at night. OK, so usually I get only about halfway through it before Mr. Sandman knocks me out, but still.
Probably next would be the Beatles’ Abbey Road; it was the second album I ever bought (after Grand Funk Railroad’s ‘Closer to Home’, don’t judge me) and I’m sure I listened to it hundreds of times (on vinyl!), with a twenty-year or so gap before repurchasing it on CD.
Seconding previous suggestions of Revolver, Moondance, and Blood on the Tracks; have all of these on vinyl, cassette & disc. But my top 2 are London Calling and Exile on Main Street.
Obviously, numbers approximate.
But I could probably add Graceland, Wish You Were Here, Very Best of the Eagles, Best of Leonard Cohen, Negotiations and Love Songs, Morphine, a couple Pixies albums, Heart of Gold, Day for Night, Joshua Tree, several REM albums…
I vote that Best Of and Greatest Hits albums don’t count. 
Queensryche, Operation Mindcrime
Fair enough. Just being honest. “Best of” isn’t really the same, but certainly a forerunner of “three MP3s from their entire discography”.