And by own, I mean either have a physical copy or bought digitally. I'm picking the girls who most of us stan for.
You can specify which ones too, and if you have a physical or digital copy.

Madonna
Lady Gaga
Janet Jackson
Kylie Minogue
Beyonce
Rihanna
Britney Spears
Avril Lavigne
Kelly Clarkson
Björk
Katy Perry
Nicki Minaj
Mariah Carey
Whitney Houston

Rihanna, Britney, Nicki & Katy - all their albums (physically)

Beyonce - B'day
Lady Gaga - The Fame & The Fame Monster

Madonna- MDNA
Gaga- The Fame Monster Deluxe, Born This Way, Gaga Holiday EP
Beyoncé- 4
Rihanna- Good Girl Gone Bad, Rated R

Those are all purchased digital copies.

I also have Whitney's self titled, physical.

Gaga, & Katy = All
Beyoncé = IASF & 4
Britney = Femme Fatale
Nicki = All
Rihanna = GGGB, RR, LOUD, TTT

Wait! I thought it said only physically. Well I bought all of these physically. I have all of their albums downloaded digitally as well.

Madonna - All twelve studio albums, Immaculate Collection, Rain, Erotica, Bedtime Story, Frozen, Ray of Light, Music, Vogue, Don't Tell Me, American Life, Nothing Fails, and Hung Up singles, Sticky & Sweet Tour, Confessions Tour, I'm going to Tell You a secret all physically. Celebration, Revolver singles, and Evita digitally.

Lady Gaga - The Fame/Fame Monster deluxe digitally

Janet Jackson - Control, Rhythm Nation, Janet. The Velvet Rope, All For You, and Discipline physically.

Kylie Minogue - Impossible Princess, Fever, and Body Language physically. Her stuff is hard to find.

Beyonce - B'day physically.

Britney Spears - Blackout physically.

Björk - Debut, Post, Homogenic, Vespertine, Medulla, and Biophilia physically.

Rihanna - All
Lady Gaga - All
Beyonce - DIL/IASF
Briney Spears - ITZ/BO
Madonna - COADF

Lady Gaga: TF and TFM
Beyonce: IASF
Rihanna: GGGB, Loud and TTT
Kelly Clarkson: All of her albums
Katy Perry: TD
Mariah Carey: EOM

Lady Gaga-TF/TFM/BTW (Deluxe)
Beyonce-DIL/B'Day/IASF (Deluxe)/4 (Deluxe)
Rihanna-AGLM/GGGB/Rated R/Loud/TTT
Britney Spears-ITZ/Blackout/FF
Nicki Minaj-PF/PF:RR (Deluxe)
Mariah Carey-Greatest Hits/The Remixes/Charmbracelet/TEOM/E=MC2

Gaga: All
Britney: All
Rihanna: GGGB:Reloaded
Beyoncé: All

I don't really own that many albums

Madonna - Celebration
Lady Gaga - ALL
Janet Jackson - none
Kylie Minogue - Aphrodite
Beyonce - I Am
Rihanna - ALL
Britney Spears - Femme Fatale
Avril Lavigne - none
Kelly Clarkson - All I Ever Wanted
Björk - Volta
Katy Perry - ALL
Nicki Minaj - ALL
Mariah Carey - Music Box
Whitney Houston - Greatest Hits

Madonna: Erotica, Bedtime Stories, Ray of Light, Music
Lady Gaga: The Fame, The Fame Monster, Born This Way
Janet Jackson: none
Kylie Minogue: none
Beyonce: Dangerously in Love, B'Day, I Am.... Sasha Fierce, 4
Rihanna: A Girl Like Me, Good Girl Gone Bad, Rated R, LOUD, Talk That Talk
Britney Spears (legally) Britney, In The Zone, Circus, Femme Fatale
Avril Lavigne: Let Go, Under My Skin, The Best Damn Thing, Goodbye Lullaby
Kelly Clarkson - none
Bjork - none
Katy Perry - One of the Boys, Teenage Dream
Nicki Minaj - Pink Friday, Roman Reloaded
Mariah Carey - Daydream, Butterfly, The Emancipation of Mimi
Whitney Houston - Whitney Houston, Whitney, Greatest Hits

Madonna: Hard Candy, MDNA
Lady Gaga: The Fame, The Fame Monster, Born This Way
Janet Jackson: None
Kylie Minogue: None
Beyoncé: None
Rihanna: A Girl Like Me, Good Girl Gone Bad, Loud, Talk That Talk
Britney Spears: Oops!... I Did It Again, Britney, In The Zone, Greatest Hits: My Prerogative, Blackout, Circus, Femme Fatale
Avril Lavigne: None
Kelly Clarkson: None
Bjork: None
Katy Perry: Teenage Dream
Nicki Minaj: None
Mariah Carey: None
Whitney Houston: None

All of them are physical copies

Madonna: Celebration
Lady Gaga: The Fame, The Fame Monster, Born This Way, A Very Gaga Christmas, and both remix.
Janet Jackson: None
Kylie Minogue: None
Beyonce: B'Day, I Am... Sasha Fierce, 4.
Rihanna: GGGB:R, Loud, Talk That Talk
Britney Spears: The Singles Collection, Femme Fatale
Avril Lavigne: None
Kelly Clarkson: Stronger
Björk: None
Katy Perry: One of the Boys, Teenage Dream: The Complete Confection
Nicki Minaj: Pink Friday, PF:RR
Mariah Carey: None (This will change, today)
Whitney Houston: None

Kelly Clarkson - ALL
Lady Gaga - The Fame and The Fame Monster
Rihanna - Rated R

Lady Gaga: The Fame, The Fame Monster, Born This Way, Cherytree Sessions EP, A Very Gaga Holiday
Rihanna: Rated R, Talk That Talk
Britney Spears: In The Zone, Blackout, Femme Fatale
Katy Perry: Teenage Dream

None of them.