Bruce Springsteen - all albums.
Billy Joel - all albums.
John Mellencamp - all albums.
Weird Al Yankovic, up to the point where I didn’t recognize any of his parodies anymore.
Jim Croce - his three hit albums before he died.
Boston - I stuck with them through Corporate America. I also bought Barry Goudreau’s album, Orion the Hunter, and Return to Zero (Goudreau & Brad Delp).
Nerf Herder - all albums.
The Presidents of the United States of America - all albums.
I believe my list is:
Alice in Chains
Stone Temple Pilots
Nirvana
Soulfly
Within Temptation
Nightwish
Kamelot
Violent Femmes, since they’ve had so few, and I appreciate their style. I’m generally not an album buyer. Thinking back, I may have the entire catalogue of a couple of 80’s alternative bands…
Working on Dylan
Tom Waits
Pink Floyd
The Police
Brandi Carlile
Tool (excluding the most recent)
Audioslave
Rage Against the Machine
The Beatles. The Dead, except for a couple of stinkers. Dylan, ditto.
I own multiple copies of all of Gustav Mahler’s symphonies, plus the song cycles.
Carly Simon
Spirit
The Velvet Underground - not just the four studio albums, but all the posthumous albums, box sets, important bootlegs, and seventy or so solo records released until about 2005.
Pink Floyd - I have all the studio and live albums (but not the super deluxe editions) up until that instrumental record that came out a few years ago.
The Byrds
The Beatles and John Lennon (not every outtake album, but the basic catalogues)
Captain Beefheart - he’s another one with a messy discography, but I have all the records released during the time he was working as a musician.
Lana Del Rey (the only current artist I’m obsessively collecting)
Led Zeppelin - I bought the box sets, which had almost all the songs they issued, then I ripped them to my PC and rearranged the tracks into their original album order, each album in its own folder.
Jim Croce
Bread
CCR
Emmylou Harris, six albums starting with Pieces of the Sky through Roses in the Snow. I’ve bought a couple more. White Shoes is a great album from 1983. I haven’t tried buying everything she’s released.
Bonnie Raitt late 80’s / early 90’s hit albums, Nick of Time, Luck of the Draw, Longing in Their Hearts. Her older stuff from the 70’s is a different style.
Kyuss - 4 studio albums and a split with Queens of the Stone Age.
Thee are a few more that almost count, but I pretty much stopped buying albums when streaming came around:
Fu Manchu
Godflesh
Electric Wizard
Got ‘em all:
[ul]
[li]La’s (one album)[/li][li]David & David (one)[/li][li]Sundays (three)[/li][li]Smiths (four)[/li][li]Keane (four *)[/li][/ul]
Collected all up to a point, then stopped dead:
[ul]
[li]Radiohead (through Amnesiac)[/li][li]REM (through Monster)[/li][li]Bruce Springsteen (through Lucky Town/Human Touch)[/li][/ul]
(*) Looks like Keane released a new CD a few days ago after a seven-year break. I guess I’ll have to go pick that one up. 
I have three, all of which I follow on social media so that I can be notified when a new album comes out and mark my calendar for when albums are released (and sometimes to pre-order):
- Hanson
- Andrew McMahon in all his different incarnations (Andrew McMahon in the Wilderness, Jack’s Mannequin, Something Corporate)
- Old Dominion
If you buy no other Springsteen album, I wholeheartedly recommend you get The Rising (2002), his first album in like 18 years with the full E Street Band. It was written partly in response to 9/11. It’s a mixture of introspective songs, light optimistic tunes, and flat-out rock ‘n’ roll party songs. There’s not a bad song on the album.
Janelle Monae
Roxy Music, up to the break-up after Siren, including Viva! (the live album) and Manifesto (the first album back together, which persuaded me to stop).
David Bowie, up to Heroes (except Pin-ups, the album of covers)
John Cale, up to Honi Soit. Hell’s teeth, I even bought Church of Anthrax. I kinda got less obsessive as I got older.
j
I’m a musician and collector so I have a massive collection.
Bands I can say that I have ALL their recordings?
The Beatles
John Lennon
Led Zeppelin
AC/DC
Iron Maiden
Rush
Kiss
Pink Floyd - up to Division Bell, I think
Queen
Eagles
Don Henley
Tesla
The Ramones
Triumph
Beastie Boys
Black Sabbath (up to Live Evil)
Ozzy Osbourne
Collective Soul
Kim Mitchell
The Kinks
Megadeth
Men at Work/Colin Hay
James Taylor
Gordon Lightfoot
The Tragically Hip
GnR
Jack Johnson
Blue Rodeo
NOFX
Goldfinger
Van Halen
Sammy Hagar - Solo/Van Hagar/& the Waboritas/Chickenfoot/The Circle, etc.
John Cougar Mellencamp
Sublime
There’s also a bunch of artist that I have most of their stuff, like The Stones, Bowie, Judas Priest, etc…
Rory Gallagher. There wasn’t a Rory Gallagher recording that I found that I didn’t buy. Eventually, I went through the catalog one by one and made sure I didn’t miss any. Not a one of them was a disappointment, either!
Forgot to mention, I have all of the Eagles’ studio albums, plus Hell Freezes Over. I haven’t bothered with their greatest hits albums.
The Beatles
Led Zeppelin
Guns and Roses (original line-up)
Metallica (except maybe the newest ones)
Hank Williams III
Nirvana
Jamey Johnson
Electric Wizard
Coheed and Cambria
I know they’re not a household name - but they are my favorite band. They’ve released **9 studio albums **and I’ve got them all. Also have some live material and have have lost track of the number of times I’ve seen them in concert.