The Church’s The Blurred Crusade. [Linking to proselytize for them, natch. :D]
I rarely listen to full albums anymore (prefer to play most of my music on random shuffle mode), but it would definitely be at the top all time. Has the most hooks of any album I have ever heard, most powerful too.
Inside Dave Van Ronk
“Pearl”, Janis Joplin tied with “Layla and Other Assorted Love Songs”, Derek and the Dominos (yes, all four sides)
Like Procrustus, I don’t listen to albums from start to finish. Or very rarely.
However, I used to listen to Björk’ Debut very often at night when I lived in London. 50 times perhaps.
Next, would be Bon Jovi’s Slippery When Wet. I listened to it a lot when it came out (I was 12).
Five come to mind immediately, each with, I would guess, well over a hundred spins.
Listed in order:
Elvis Costello - Imperial Bedroom
Paul Simon - Graceland
The Beatles - Abbey Road
Joni Mitchell - Court and Spark
Lou Reed - Transformer
mmm
Most likely “Murmur” by R.E.M. - thousands of times over the decades. 
At a quick estimation I’d guess Something To Write Home About by the Get Up Kids. Just on my PC it’s been played 40 times, plus I’ve listened to it a bunch on CD and vinyl. Other albums on my PC that have more plays are newer and thus haven’t had as many spins.
Rounding out my top 4 are probably The Wall, 3 Cheers for Sweet Revenge by MCR, and A Fever You Can’t Sweat out by Panic! at the Disco.
The Yes Album
Probably Springsteen’s The Wild, the Innocent & the E Street Shuffle.
I’ve had some albums for 60 years, in one medium or another. So it would have to be something I’ve had for at least several decades. Possibly the most played would be the Vladimir Ashkenazy recording of Rachmaninoff’s Piano Concerto #3. Or maybe Chopin’s Etudes, played by Maurizio Pollini. Or Verdi’s Rigoletto, with Sutherland and Pavarotti.
I’d have to say Don Henley’s End of the Innocence. This was back in the days of cassette tapes and I literally wore 3 cassettes out. No idea how many times I listened to it. Loved every song on it and still do.
The Black Album and Strawberries
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Either Jagged Little Pill or Big Boi and Dre Present. . . Outkast because I had them on my playlist at work when I started at a job that was very repetitive and we were allowed headphones. The advent of digital playlists really put these guys over. A close third would probably be either Maroon 5’s Songs About Jane and Fleetwood Mac’s Rumors. Maybe even The Best of Earth Wind and Fire.
How many times have I listened? Countless. We’re talking more than 10 years, 8 hours a day.
Who’s Next, Born to Run, Blood on the Tracks. Take your pick.
Linda Ronstadt’s Heart Like a Wheel. I bought the vinyl twice, a cassette, and eventually a CD.
Probably one of my Seventies favorites like “Who’s Next” or Yes’ “Close to the Edge.” Or MAYBE “Led Zeppelin II.”
I’ve only got a few albums that have been companion pieces to my life. By that I mean that I’ll happily stick one of them on and listen all the way through. There’s hundreds/thousands of songs, but not that many albums that I’ve liked in their entirety from the first time I heard them to me being nearly 50. They are comfort albums at this stage, I guess.
The Velvet Underground and Nico
The Velvet Underground
Cocteau Twins - Head Over Heels
Cocteau Twins - Treasure
The Smiths - Hatful of Hollow
I’m sure it is no coincidence that I first heard these as a teen.
Elvis Costello- My aim is true
REM- Green
The Smiths- Strangeways here we come
And in recent years it’s Mellow Candle’s hidden gem of progressive folk “Swaddling Songs”
Boy, that’s tough.
December, 1980 and I was stuck in Corvallis for five moths in between college years at UCSB and I just about wore out Linda Ronstadt’s *Simple Dreams *on vinyl.
Or maybe Pink Floyd’s The Wall. 1983, back in college and the theme and music on cassette were the backdrop to some pretty tough times.
I don’t imagine I’ve ever listened to music so compulsively as during those two periods…