The only persons discog I legally own is Eminem's. I have own all 7 of his studio albums + the re-release of Relapse + his Greatest hits + the BME EP and the Re-Up if those can be counted as part of his discography.
Avril Lavigne
Beyoncé
Billy Talent
Britney Spears
Cascada
Christina Aguilera
Demi Lovato
Fall Out Boy
Girls Aloud
Kelly Clarkson
Lady Gaga
Maroon 5
Miley Cyrus
P!nk
Panic At The Disco
Paramore
Rihanna
Selena Gomez
Shakira
Simple Plan
arcade fire, beyoncé, björk, britney, ciara, fiona, goldfrapp, kate, kelis, leona, los campesinos!, madonna, patrick, rihanna, shakira, st. vincent, the cure, xtina
i'm missing like one each of cher, david, kylie, mariah, nina, qotsa and sia's albums
Taylor Swift
Miranda Lambert
The Dixie Chicks
Avril Lavigne
Carrie Underwood
Mariah Carey
Jewel
Kellie Pickler
Sara Evans
Demi Lovato
Christina Aguilera
Gavin Degraw
Leona Lewis
Dierks Bentley
Lady Antebellum
Shania Twain
The Used
If you mean legally, then only Gaga
-The Fame
-The Fame Monster
-Born This Way
-ARTPOP
-A Very Gaga Holiday
Plus Red & Blue + the remix album + the Cherrytree thing illegally.
If you mean illegally, then a lot of people because I just download artist's discographies rather than separate albums.