Deus Ex. It’s a really good game, but I got tired of running around Shanghai not knowing where I was supposed to go, and just stopped playing it.
If you play it long enough, you get to smash Montreal and Toronto.
A couple have mentioned Grim Fandango. I loved that game and did finish it. Can’t remember how it went though. It’s the last great game of that type I ever played. We need more.
PC: Nethack and all the others in its genre that I’ve played. Possibly Morrowind, not sure if I’ll ever finish it.
Commodore 64: there’s so many! Brian Bloodaxe, Underwolrde, Hitchhiker’s Guide to the Galaxy, Strike Fleet, some Gauntlet-style dungeon game from Electronic Arts which featured a construction-set program on side two (can’t remember the title)…
Never got around to finishing Doom III but I blame my system for that. It was too dang choppy (yes, I had the graphics turned down). I never managed to finish Zelda: A Link to the Past either (for GBA). Oh yeah, the original Thief I never finished. I’d really like to get the new one and try again though 
Star Ocean: The Second Story for PSX. I’d always start it and then my bro would take his 'station to school. Now that I have it, I still haven’t finished the stupid game.
Also, FFIX. I was on the fourth disk, and somebody saved over my game. Pissed me off enough I never started a new game.
I don’t think I finished any of the campaigns in Age of Empires II. Got to the last scenarios of quite a few, but never finished a single one.
Hiebram
I never finished Planescape: Torment, although I came very close.
Never beat the main story line for any of the GTA games. In San Andreas, I never even got out of the starting city.
Never finished Knights of the Old Republic. Got to the big plot twist, thought, “Oh, that’s neat!” and never played it again.
Plenty more where that came from, once I can think of them.
I never finished Mist 3.
The only video game that I was get all the way through was Dragon’s Lair.
I always wanted to see if there was an end to the old Intellivision “Pitfall” game.
For the SegaCD: Darkwizard- There’s this one battle where you need one particular summon spell for- I know where it is, have all the items (4 or 5 times), and I just can’t trigger her appearance. So, while I can usually nuke everything from a day’s march, I can’t progress.
Legend of Zelda, OoT- just stopped playing.
You didn’t miss much. The boys went back to San Dimas and had a jam with the princess babes. Da end.
I’m one of those that it’s easier to count the ones I’ve finished than ones I haven’t. But some of my favorite ones I never made it through are:
Wing Commander I & II - Loved the games but never made it through.
Final Fantasy 7 - Just too darn long.
Banjo & Kazooie - Made it to the very end, I mean the very end but I couldn’t get past the final confrontation with the witch.
Warcraft 3 - Again, made it to the very end but couldn’t win the last battle. Poo. 
Legend of Zelda: Ocarina of Time - I don’t know how many times I started this game but for one reason or another, I never got through it.
Super Mario 64 - Great game, but just couldn’t finish it.
Civilization and Railroad Tycoon - played different versions of both games but never made it through any of them.
And just to make myself feel better, the ones I did finish:
Halo
Knights of the Old Repbulic
Return to Castle Wolfenstein
and Call of Duty
I bought an Xbox about a year ago and I said, “By gum, I’m goin’ to get my money’s worth this time!” So I worked my way through these but had to stop because work got too busy. Maybe during the summer.
I am really, relaly bad about actually finishing games, whether I like them or not. Often, I’ll get stuck somewhere and give up. Sometimes I go back later, sometimes not. I have yet to ever actually finish a Final Fantasy game, for example. I love them all, and I’m really, really close to the end on 9 (have the gamesave still, but the game isn’t mine), but I’ve yet to actually finish one, and I’ve played every number up to X (Well, X-2 but I don’t like that one much) and both Tactics. Other games:
Golden Sun 2
Neverwinter Nights–On the second chapter
Pokemon Yellow (Shut up. I think it was just boring since I’d already finished Red twice)
Aidyn Chronicles–Actually never got far. Rented it once but couldn’t save the game, finally got a copy and a save cartidge, but our TV sucks and I can’t see a thing
MagiNation
Bahamut Lagoon
Star Ocean (The original)
Chrono Trigger
Chrono Cross
Secret of Mana & Seiken Densetsu 3
Hey You, Pikachu (I repeat, shut up. I swear the thing can’t decipher my accent)
Some of these I may go back and finish someday, some I doubt I’ll ever bother with again.
There are people who actually have time to finish games? I’d have an easier time naming games that I have finished; I can only think of 4 or 5 at the moment. Hell, I’ve worked on games that I’ve never played all the way through.
The ones that I really liked, but never finished:
Half-Life: got to a point where my only save game had me with 5 health and pinned down in front of a turret that would kill me instantly. I tried recently to play through it again, after finishing HL2, but it’s just too ugly now, and doesn’t run well on my machine.
Space Channel 5: Granted, it’s not anything more than a flashy version of Simon, but it gets 1000 style points. I got to the second-to-last level (after unlocking Space Michael Jackson) and completely lost interest.
Freedom Force: Still love the style, but hate the game (I keep buying RTS games, but am terrible at them and don’t really enjoy them). I only got about four levels in. Bought the sequel anyway, only got through a single level before I lost interest.
Baldur’s Gate (PC): Got to the next-to-the-last chapter, never bothered to finish it. I started the sequel, and that ruined the ending anyway.
No One Lives Forever: Got about halfway through, became annoyed at the clumsily implemented stealth missions, gave up.
Advance Wars: Spent about an hour on the very last mission before I got slaughtered, never tried it again.
Seaman: Drained the tank, he turned into an amphibian, I went on vacation for a week, forgot to set the clock back, he died of starvation.
Halo: I reckon I got about halfway through, something else came up and I didn’t touch my Xbox again until Halo 2 came out.
Crimson Skies: Either version. The PC version was so cool but pissed me off on the gyrocopter level; the Xbox version I still intend to finish someday.
X-Wing, and TIE Fighter: Both great games, but I suck at flight simulators and didn’t have the patience to finish either.
Man, I loved that game. I did finally finish it with the help of my roommate, and at the risk of spoiling it for you, I’ll say that it was more than a little anticlimactic. IIRC, though, they flash up a certificate saying that you won, so you can print it out and presumably frame it.
The Monkey Island that was around about a decade ago. I got all three tshirts and then didn’t know what else to do.
Hacker. I never did beat the little bugger. I have it with an emulator, but I still get stuck in exactly the same places. I think it’s impossible. The only walkthroughs available for it on the Net are wrong.
Final Fantasy X-2. No real reason. I’m just about to go to the area where the final boss is. I think I wanted to level up and then I got really bored after fighting for like 20 hours.
Chaos Bleeds. The zoo level kicked my ass and I hate games that kick my ass. They aren’t fun, just stupid and frustrating, so I quit.
My husband hasn’t finished Final Fantasy IX, Final Fantasy X, or Final Fantasy X-2, and I think he’s still working on Kingdom Hearts and a handful of other games I don’t remember. I told him he couldn’t buy more until he actually finishes one, just one, of the others.
Marathon I got stuck in this one area and then my Mac died.
Neverwinter Nights I got to a certain point and had my PC crash and burn and I had to do a reinstall so I lost my saved game. So I started over and after a while the computer died again, that time permanently. My laptop can’t handle it so I can’t play it now. It makes me a little nervous about installing it on a new computer, perhaps it’s jinxed?
You’re not missing much on the ending with X-2. Though, aparently if I had completed more of the game I would have had a different ending. I didn’t care enough about the game to try it over. Ride the Shoopuff!!!
Anyway, this list brings back a lot of memories. One game I remember distinctly not beating is X-COM 2 (I think it was 2… the one under water). I played it for a while but the dificulty curve was too much; it just got impossible. I liked it up until it was ridiculously hard.
Also, to go back even farther, Kings Quest III and IV. III in particular was hella tough.
The first thing that came to mind was Ultima V on the Apple IIe. I defeated all three Shadowlords, collected all the artifacts I needed, and all that was left was to rescue Lord British himself, but I could never figure out how to get to him. I found him trapped in the underworld in some sort of crystal but I couldn’t reach him, there was a barrier in the way. I could have gotten to him from the other side, but I never found a path that led there.
Then there are games that are somewhat open ended. Take Gran Turismo, for example. After I won the F1 championship series, what fun was it to go back and race in the Volkswagen Lupo Cup? Or any of the Grand Theft Auto games, where I finished the main storyline but not all of the little side missions.