Gotta Have it All!

[Bogey]

Peter Gabriel

Good call.  had all of his stuff twice and dumped it twice.  I might go back to that well again....

Dire Straits anyone?

[Bulldog]

What artists (with a sizable musical output) do you like so much that you need to have everything they've ever recorded?

None.  I loved groups like the Beatles, Stones, Dire Straits etc. but never had interest in owning "everything".

[TheGSMoeller]

Tom Waits. 

[71 dB]

Good to have nearly everything:

S'Express
The Beatmasters
The Prodigy
Jonny L
Autechre
Tangerine Dream
King Crimson
Carly Simon

[Leon]

Also,

Thelonious Monk
Modern Jazz Quartet
Bill Evans
Andrew Hill
Mose Allison
Dr. John
Professor Longhair
(Don't laugh) Stephen Bishop
James Taylor

[Scion7]

Many.  But only because their entire output was worthwhile - that doesn't mean that other artists didn't surpass them in the majority, but those other artists may have had period(s) of apprenticeship that weren't notable, or declined during some period(s) - Charles Mingus, for instance - much of his music is part of the pinnacles of Jazz.  I have everything of his from 1957-on, but not stuff earlier than that.

Eric Dolphy and Family are two that I grabbed everything by.