I don't think I own an entire discography by any band. Even though I love Pink Floyd, I still don't have the rare single 'Point Me At The Sky' although getting hold of the music for 'La Carrera Panamericana' was a bonus.

Close but no cigar:

Fela Kuti (16 releases)
Porcupine Tree (28 releases)
65daysofstatic (6 releases)
Black Sabbath (14 releases)
Ben Harper (8 releases)
Carter (the unstoppable sex machine) (5 releases)
Clannad (10 releases)
Crass (6 releases)
David Bowie (18 releases)
Death (6 releases)
Dreadzone (6 releases)
Elliott Smith (8 releases)
Fields Of The Nephilim (11 releases)
Hawkwind (12 releases)
Iron Maiden (11 releases)
John Martyn (10 releases)
Kate Bush (8 releases)
Megadeth (9 releases)
Pink Floyd (17 releases)
Queen (24 releases)-still never ever heard a full album by them.
Simon and Garfunkel (6 releases)
Slayer (15 releases)
The Beatles (10 releases)
The Clash (15 releases)
The Cure (11 releases)
The Notwist (6 releases)
Thievery Corporation (10 releases)
Underworld (6 releases)
Ministry (22 releases)
My Bloody Valentine (12 releases)

I have:
The Beatles (all studio albums, singles, b-sides and EPs)
Velvet Underground (all four studio albums; I don't count the last one)
The White Stripes (all six studio albums)

And a few bands with just one or two albums.

Most the bands I like have very large discographies, and I can't be bothered to get their mediocre stuff just for the sake of it.

David Bowie
Elvis Costello
Nick Cave
New Order
Talking Heads
Supergrass
Blur
Paul Weller

Led Zepplin
Elcho....they're an awesome indie band from the UK that broke up last year -__- So, not hard to collect all their stuff

PM'd.

Joy division 2 albums 3 compilations
The smiths all studio plus a couple of live and some bootlegs
The cure all albums apart from the new one yet to be released, compilations, a few eps, and many many bootlegs.
Radiohead all studio albums, plus eps and bootlegs.
The beatles all albums and some compilations
Bloc Party all albums plus remix album
Bright eyes all albums
Brittish india both albums
Yeah Yeah yeahs two albums and the is is ep
CSS two albums and ep
Cut copy albums
Daft punk albums and alive
Death Cab For Cutie all albums
Dirty pretty things - two albums
the doors - everything
The dresden dolls- three albums
Elliot smith - all albums 2 compilations
Gold frapp - all albums
The birthday party - everything
Kings of leon - all four
The strokes - 3
The libertines - both
The mars volta - all albums
Nirvana - all albums
Ratatat - all albums and a couple of mixtapes
The presets - both studio albums, a couple of eps and mixtapes
Sufjan stevens - albums
The white stripes - albums
David bowie - albums and a few compilations

heaps of other artists that just are recent or only ever released one or two albums and i have neglected to include

Queen

I'm working on The Beatles.

Almosts:
Porcupine Tree
Johnny Cash
Anberlin
Lagwagon
Placebo

Probably only Jimi Hendrix, although chances are there are some compilations that I'm missing.

The three Experience albums
Band of Gypsys
Live at Montery
Kiss the Sky compilation

Have you heard/do you have Resolve?

If not, it's a very good listen.