For me, it depends.
Say, for instance, we're talking about HANSON. The only greatest hits CD released of their music merely contains the early stuff from their first two proper albums, and also from quickly cut Christmas and early live albums. But Hanson's best stuff has all been recorded after they were dropped from their major label and went independent. So the only real option is to get the individual albums. All of them. Of course, that's what any real fan would do anyway.
Other music where I've got to have all of the individual albums would be: Suede, Oasis, Radiohead, Teenage Fanclub, the Oak Ridge Boys, Eric Johnson, and quite a lot more.
Some bands I do a little of both: A hits compilation CD plus many of the individual albums. For example, the BEACH BOYS I've got at least 15 of their individual albums, but some of the very earliest songs (and some of the later songs from 1980 and newer) I only have on the greatest hits CDs and that's OK. Other bands that I do this with include Cheap Trick, CSN&Y, Bowie, Prince, Barenaked Ladies, Tears for Fears, Nirvana, Pearl Jam, Dwight Yoakam, Sam Cooke, the Beatles and more.
For other music, I'm totally fine with greatest hits compilations and no individual albums whatsoever. I especially like those 2-disc sets that Sony Legacy puts out. But there are other good ones, too. Greatest Hits albums that I like and feel complete enough for me to never buy any of the individual albums are as follows:
1. Billy Joel - "The Essential" (the only thing missing is "Pressure" for some reason; though, it is available on the DVD of the same name)
2. Hall & Oates - "Ultimate" (later released as "The Essential"
3. The Byrds - "The Essential"
4. The Temptations Box Set "Emperor's Of Soul"
5. Edit: David Ruffin's two hits collections from Hippo Select (Edit: nevermind... I think those are mostly his individual albums in guise of a comp)
6. Eddie Kendick'(s)' - "The Ultimate Collection"
7. Willie Nelson - "The Essential"
8. Sammy Hagar - Essential Red Collection
9. Sam Cooke - "Portrait of a Legend" is flippin awesome and I'll post here for recommending. (But with that said... I actually supplemented it with a two additional comp albums, some live albums, a single proper album, and one of his Soul Stirrer's-era comp albums.)
And I'm quite happy to simply have only Greatest Hits collections (for the most part) of the following: The O'Jays, Average White Band, Paul McCartney & Wings (Wingspan), Alice In Chains, Randy Travis, the Doors, Elvis Costello, ELO, the Eagles, ABBA, Men at Work, Moody Blues, Seal, Skid Row, Supergrass, U2, warrant, Bangles, The Band, the Bee Gees, Black Crowes, Boston, Springsteen, Soundgarden, the Cars, EWF, Faith No More, Hollies, Huey Lewis, Hendrix, John Mellencamp, Mamas and the Papas, the Rolling Stones... I could go on and on.