Although some will decry the existence of best-ofs, they absolutely serve a purpose. They can appeal to the casual fan or the diehard colector who must have everything, or they can get someone started on a long love affair with an artists's music.
Although many best-ofs were painfully short, there are so many memorable ones from when I grew up. Many of these albums were where I got into an artists for the first time, and quite a few had cool album covers. I'm just going off the top of my head at some of the classic best-ofs (official releases only-no bogus releases by crappy labels) from the good 'ol days here:

Deep Purple-Deepest Purple:The Very Best Of Deep Purple
Rolling Stones-Made In The Shade
Through The Past Darkly
Hot Rocks
Sucking In The 70s
John Lennon-Shaved Fish
Pink Floyd-Relics
A Collection Of Great Dance Songs
Works
Black Sabbath-We Sold Our Soul For Rock And Roll
Foreigner-Records
The Cure-Standing On A Beach:The Singles
Echo & Bunnymen-Songs To Learn And Sing
CSN&Y-So Far
Journey-Greatest Hits
Kiss-Double Platinum (even with the remixes)
Eagles-Their Greatest Hits 1971-75 (even though nothing was released until 1972!)
Doobie Brothers-Best Of The Doobies
Steely Dan-Greatest Hits
Thin Lizzy-The Adventures Of Thin Lizzy
Ultravox-The Collection
Chicago-Chicago IX Greatest Hits Vol. 1
Jethro Tull-MU The Best Of Jethro Tull
Moody Blues-This Is The Moody Blues
Bad Company-10 From 6

Deepest Purple was my introduction to Deep Purple. At the time, the band wasn't together, but I'd read an article about them in Hit Parader and was intrigued. It had nearly an hour of music, which was something you didn't see on most LPs.

Bob Dylan's Greatest Hits Vol II ---- My first Dylan album. It will always be on my list of Top Albums, somewhere.

A few that stand up as great albums in and of themselves:

Cat Stevens: Greatest Hits
Sly & the Family Stone: Greatest Hits
New Order: Substance (even though it's mostly 12" remixes)
Bob Marley: Legend
Beatles: Hey Jude, 62-66, 67-70
The Who: Meaty Beaty Big and Bouncy

Simon & Garfunkel - Greatest Hits

I like the alternate live versions on it.

Chuck Berry's Golden Decade (later vol 2 & 3 followed)
Mothermania - F Zappa & the Mothers of Invention
The Best of the Lovin' Spoonful - The Lovin' Spoonful

3 that quickly come to mind for me are Poison, Cat Stevens & Don Henley. All 3 artists have a few songs I like alot, & NO complete albums I want. I especially recommend the Henley.

I can see 6 reasons for buying a best of:

1. An artist that only has a few songs I like
2. Lack of money
3. To check out an artist I'm not familiar with
4. Quality remastering not available elsewhere
5. Rarities not available elsewhere
6. Completeness (i.e. box sets)

Jimbo:
absolutely to New Order and The Who. That Who album had many singles previously unavailable on any albums.

I suppose the original Wings Greatest Hits too. It had enough of the essentials, Mull of Kintyre, and stopped before the '80's.

I've never been a big solo Paul fan, so it has all I need.

Hall & Oates -- Rock & Soul, Part One.

Unlike some collections, the "new" tracks on this were top notch and actually became hits. I hated it when they changed the cover to a generic live shot, though. Recently reissued with the original cover (finally!) and a couple of bonus tracks.

I agree especially on Sly (always been my fav Sly release), Cat Stevens & Hey Jude.

(Just my personal opinion) I don't really consider Hey Jude to be a greatest hits album....(Old Brown Shoe, a greatest hit?) ......just an album that rounded up most of the rest of what was only singles in the US at the time.

It is one of my favortite Beatles albums.

Yes to Meaty Beaty Big & Bouncy, and Sly & The Family Stone too.

The 2 CD import of Stevie Wonder's Difinitive Collection pretty much has every essential song you want to hear.
(aside that Talinkg Book through Songs In The Key of Life are essential).

+1 on the Wings and Who collections. And Kiss and Deep Purple.

Add, for me, "Elton John's Greatest Hits Vol. 1" and the first two Johnny Cash "Sun Records" recordings.

There may be more in my collection as well, but I'm too lazy to look right now.

Alan

I'll always have a soft spot for Aerosmith's "Greatest Hits", slight as it was, it was a great collection.

Cream's "Strange Brew" was another tape I wore out as a teen in the 80s.

I don't know if these are still in print or if they have been superseded by expanded 80-minute best-ofs, but these two were great collections.

The original Chicago's Greatest Hits was wonderful and played often.

Oh yeah, The Kinks' The Kink Kronikles!!!

The Rolling Stones, "Hot Rocks 1964-1971".

Tom Petty's has virtually every worthwhile Petty track (minus the Wilburys material) and also produced a kick ass single.

That was a great one, especially the SACD remaster.

"Hot Rocks" takes me back to when I was a kid and had a lot less money to spend on music--my attitude towards greatest hits compilations was considerably diffferent then than it is now. (I should remember this fact more often)

This plus The Who: Meaty Beaty Big & Bouncy

I'll go with:

History -- America
IX -- Chicago
Deepest Purple -- Deep Purple
Greatest Hits -- Cat Stevens
M.U. -- Jethro Tull
Repeat -- Jethro Tull
Chronicle -- CCR