The only dvds that I've double dipped on and I'm keeping both versions are:

The Lord of the Rings Trilogy
Terminator 2 Ultimate & Extreme

What else has anyone kept?

Donald Darko

I'll be keeping my Man With No Name trilogy box for original audio. Also the R3 imports of Oldboy and Tale of Two Sisters, mainly for the packaging.

Evil Dead, kept the 4:3 Elite disc and have the AB WS Book of the Dead version

Doubles that I have... Of course, some are different cuts, making them different films. Strange that my SO doesn't seem to understand that.

Apocalypse Now / Apocalypse Now Redux : Two films.
The Exorcist 25th / The Exorcist TVYNS : Two films.
The Good, The Bad and The Ugly / The Good, The Bad and the Ugly SE : Two films.
Dr. Strangelove SE / Dr. Strangelove 40th SE : Goes with the Kubrick box, new transfer.
Lawrence of Arabia LE / Lawrence of Arabia Superbit : Extras, superior transfer.
Sid & Nancy CC / Sid & Nancy : Extras, anamorphic transfer.

Glengarry GlenRoss - LD has a Jack Lemmon commentary
Evil Dead - Elite LD is BLOOD RED in color
Taxi Driver - Criterion LD has unique commentary and supplements
Raging Bull - Criterion LD has unique commentary and supplements
Eyes Wide Shut - Region 3 is the uncensored version / Region 1 has interviews
Mulholland Drive - Region 3 has interviews / Region 1 has DTS
They Live - Region 2 has commentary / Region 1 has better transfer
Prince of Darkness - Region 2 has commentary / Region 1 has better transfer
Andrei Rublev - Criterion and RusCiCo editions feature different cuts
Solaris - Criterion and RusCiCo editions feature different extras
Star Wars box sets - Definitive Collection on LD has unique extras and original versions of the films sans cgi / recent DVD box has new fx and unique extras

I've kept all the original releases of the Star Trek movies. Star Trek II & VI are different versions that the collector's editions - but I've kept all of them.

Just this,Battle Royale, and FOTR for me...So far.

Saving Private Ryan - 2 disk
Black Hawk Down - 2 Disk

Anything I've double-dipped on I've gotten rid of my original verision.

I have both Daredevil DVDs, but don't consider that one.

Time Bandits: Criterion and Time Bandits: Divimax

I was considering keeping the theatrical versions of the Lord of the Rings movies along with the EEs, but instead I'm going to give them to my mom.

Edit:
I've also kept the Lawrence of Arabia: LE and Lawrence of Arabia: Superbit.

Well, it depends on what the old had and what the new one doesn't, so-to-speak.

The ones I've kept are:
T2-UE to T2-EE
[b]Daredevil-SE[b] to Daredevil-DC because the DC lacks the extensive 2nd disc special features and two different commentaries.
LOTR-TCs to LOTR-EEs for obvious and well-known reasons
oh I forgot...
Das Boot-DC to Das Boot-Superbit to Das Boot-Uncut Version. I don't have the old DC on my shelf anymore, and as a matter of fact, don't even know where it is.

And that's really about it.
I dumped my Hellboy-SE because the Hellboy-DC had everything the SE had plus more.
I didn't keep the original releases of the Matrix Trilogy because the stuff on the Ultimate Collection was so much more indepth, and the lose of fluff stuff was inconsequential.

Four times for me:
This is Spinal Tap - Criterion/MGM
Silence of the Lambs - Criterion/MGM
Clerks - Original Miramax release and Clerks X
Terminator 2 - Ultimate Edition/Extreme Edition

I kept disc 2 of the LOTR threatrical releases for the extras. I wish I would have kept the Underworld theatrical cut for the commentaries that were not carried over to the extended cut.

Didn't the Hellboy-SE have a different director's commentary?

I only have the DC / Bust version BTW.

Edit: Yes, DVD Rewind shows that the SE had an "Audio commentary by Guillermo del Toro (director) and Mike Mignola (co-executive producer) "

Hmm... I would like to listen to that, actually

As for my Double Dips I've kept:

LOTR Trilogy
T2 Extreme and Ultimate
Resident Evil
12 Monkeys - DTS and SE
Family Guy - Freakin' Sweet and the first 3 seasons
Stir of Echoes
The Matrix - Original and Special Collector's Edition Box
Alien Legacy and Alien Quadrilogy
Rocky Box Sets - Original / Re-Mastered Box Sets
Scarface - Original Release / Box Set
Silence of the Lambs - Criterion/MGM
The Fifth Element - R1 SE and R2 (Taxicab version)
Back to the Future - R1 and R4

I'm sure there are others, but those are off the top of my head

I kept the original Amadeus since the double disc didn't have the theatrical cut. I'm not happy with the fact that it's a flipper, but what can you do?

I've double dipped on Clerks, gave my original copy to a friend at work, and also dipped on LOTR and plan on keeping them. Now what's gonna happen when Star Wars comes out? Who knows.....

Echoes of others:

Amadeus / Amadeus: Director's Cut
Apocalypse Now / Apocalypse Now: Redux
Das Boot: Director's Cut / Das Boot: Original Uncut Version
Lawrence of Arabia: Limited Edition / Lawrence of Arabia: Superbit

In addition, I have all the classic Charlie Chaplin features films (City Lights, Modern Times, etc...) in both their original Image releases and the current Warner Brothers 2-Disc sets.

I've also kept seven of the original Universal "Classic Monster Collection" releases (Frankenstein, Dracula, etc..), even though the same films are in the new "Legacy" collections. (Along the same lines, I have the old Abbott and Costello Meet Frankenstein "Comedy Legends" release, though the film is included in The Best of Abbott & Costello: Volume 3.)

All 3 LOTR movies
Hellboy
Spider-Man 2 (both Superbit and SE)
Legends of the Fall (both Superbit and SE)
Daredevil (Regular & Director's Cut)
ET: Regular SE & Collector's Gift Set (I got good deals on both and the regular version has DTS while the Gift Set doesn't)
Planet of the Apes (Single Disc Version in Evolution Box Set & 2-disc SE)
Reservoir Dogs (Mr. Brown & Mr. Blonde Cover; both versions are autographed by Michael Madsen)
Rocky (SE & New Version in box set)
Terminator 2 (Ultimate & Extreme)

Now that I look at it, that's a lot more than I thought

Daredevil and Mulan...I dont know why I havent gotten rid of the Mulan yet, guess I am holding out that it will be worth something one day, even though I know it wont.

One hasn't yet been mentioned. Monster's Ball: regular and Signature Series. Different cuts, different extras, different commentaries.

The main one for me is Donnie Darko. I could see having two versions of Time Bandits as well.

As for Lord of the Rings, I far prefer the theatrical versions, but since I was able to get the extended edition trilogy as a "creative" CH order, I keep those around as well.

Only ones I've kept have been due to extras on the original version:
Saving Private Ryan
Daredevil

Planning to upgrade and keep:
Gladiator
Apollo 13

Lawrence of Arabia : SB/LE
Halloween : 2 disc LE / Divimax
Halloween 4 : LE tin / Divimax (upcoming)
Dances With Wolves : 2 disc DTS / 4 disc (korean leather package)