I have 2 of them

Panic Room Special Edition
Panic Room Superbit

Terminator 2 Extreme Edition
Terminator 2 Ultimate Edition

What about you?

The only movie I have two of is Daredevil: I have the regular version and the director's cut. I'll usually sell the older one to buy a new one in any other case.

Animatrix and Matrix (got them both again in the Matrix boxset)
Gladiator

Aren't there already a lot of double dipping threads?

Yes, but we're double dipping on the topic.

Airplane
Billy Madison
Blazing Saddles
Don't Be a Menace
Dumb and Dumber
Half Baked
Happy Gilmore
Liar Liar
Not Another Teen Movie
Something about mary
Tommy boy
Daredevil
Carlito's way
Goodfellas
Scarface
Evil dead 2

I also have 2 copies of fantastic 4. Same movie, just different sets (best buy/walmart)

No, I sold the older editions somewhere along the way.

There are several Jackie Chan, and other Chinese movies, that I have both the R1 and R0 versions of... some I even have the R2 HongKong Legends of as well

I also kept both copies of The Cube and Donnie Darko.

I kept the original Donnie Darko disc after I got the Director's Cut.

Trainspotting
Backbeat

No. I always get rid of the earlier versions before I buy the better editions later on. To me, it doesn't make any sense to own two or more copies of the same movie.

I kept the old Titanic for the trailers.

If there's a big difference in extras (like between a Criterion title and studio-released title), I'll keep both. My list:

Time Bandits
The Silence of the Lambs
This is Spinal Tap
Robocop
Toy Story
Toy Story 2
Little Shop of Horrors (Recalled and Regular)
Terminator 2: Judgment Day
Ferris Bueller's Day Off (kept the old version for the missing commentary)
Spider-Man (2-disc and Superbit, for commentary)
Lawrence of Arabia (regular and Superbit)
Lord of the Rings Trilogy (theatrical and extended for each movie)
Gladiator

I also kept the old Titanic disc for the trailer, but in a separate sleeve in the new packaging.

Naah ... only keep the one in my library.

I did the same thing. Different cuts of the movie with different music. I'm also considering keeping my old Dumb and Dumber as I've heard there is a song that is in the original thats not in the new version.

Halloween
Scarface
Goodfellas
Death Race 2000
Lord of the Rings

Gladiator: Signature and Extended
All Lord of the Rings: Theatrical and Extended
Donnie Darko: Regular and DC
Hellboy: SE and DC
Apocalypse Now: Regular and Redux
Daredevil: Regular and DC
Black Hawk Down: Superbit and 3-Disc
Underworld: Superbit and Extended

I'll only keep the original release if it's a different cut of the movie that's unavailable on the new disc, or there's a really good feature/commentary on the original.

Off the top of my head, I've got multiples of:
Clerks (original version autographed)
Daredevil
Donnie Darko
LOTR Trilogy
Hard Boiled (Criterion & Fox Lorber)
Silence of the Lambs (Criterion & MGM SE)

Ditto. I have the R2 HKL version of Chan's Miracles to view on my Oppo and projector, and the R0 Media Asia version to watch on every other player in the house.

Manhunter: UK SE which has the "theatrical cut" and extras, the Director's Cut has Mann's commentary. Then I have the Japanese disc because it's the true theatrical cut. Same case with Thief: Region 1 for the commentary and booklet and UK for the true theatrical cut. I'm a Mann purist.

Add Clerks and Mallrats to my multiple copy list. I also have The Texas Chainsaw Massacre (original) with different covers. I also still own both versions of Saw and Saw Unrated. The first one's cover was too cool to get rid of.

These I'm keeping:

The Devil's Backbone - Orig was autographed
Clerks - again, orig was autographed
The Exorcist

These I'm eventually dumping:

Jaws
Scary Movie 3
Ringu
The Frighteners
Showgirls
Demonlover
Zombie
The Texas Chainsaw Massacre
Saw

I keep upgrading certain films to the best versions available, which leaves me with duplicates; I suppose I should sell them soon.

Who Framed Roger Rabbit (original barebones & Vista Edition)
TRON (original barebones & 20th Anniversary Edition)
The Pirate Movie (regions 1 US, 2 UK, 4 AU editions)
The Amityville Horror (1979) (r1 original WS MGM release, r1 WS boxset edition)
Amityville II: The Possession (r2 UK FS, r2 DE WS, r2 UK WS SE, r1 WS boxset edition)
Amityville III/3-D (r2 UK FS, r2 DE WS, r2 UK WS SE w. 2D & 3D versions, r1 WS boxset edition)
Dolls (r0 HK, r2 DE, r1 MGM)
Super Mario Brothers (r3 HK, r1 US)
Donnie Darko (T-Cut & D-Cut)
Halloween 6 (T-Cut & P-Cut)
Star Wars Episode I (Theatrical-Cut, official r1 version, The Phantom Edit, The Phantom Re-Edit, The Phantom's New Hope)
Star Wars IV-VI (original versions, 1997 SEs, 2004 SEs)
Leprechaun 4: In Space (crap Lion's Gate & better Trimark version)
Texas Chainsaw Massacre (r2 UK 2-disc & r1 Pioneer SE)
Texas Chainsaw Massacre III (r2 DE crap version & r1 US SE)
Texas Chainsaw Massacre: A New Generation (Canadian & US cuts)
Carnival of Souls (Image & colorized versions)
Night of the Living Dead (1968) (Elite & Off-Color versions)
Jason Goes to Hell (r2 & r1 versions)
Hellrasier III: Hell on Earth (r1 fs, r2 DE Red-Version, r2 UK SE)
Burnt Offerings (r2 FS & r1 WS)
Child's Play 3 (r4 AU & r1 US)
My Name is Nobody (r1 WHAM! crap version, r2 UK, r1 Image versions)
The Tomorrow People (r1 & r2 releases)
How the Grinch Stole Christmas (r1 WS & r1 2-disc SE)
Ju-On: The Grudge (Japanese & US versions)
The Exorcist (25th Anniversary SE & TVYNS)
Exorcist: The Beginning & Dominion: Exorcist Prequel