tinyP gave me this unholy idea http://www.gog.com/forum/general/oldest_game_you_own/post43
so blame him ;p

anyway the oldest games i own i have beaten are most likely
broken sword ( 1996 )
discworld 2 ( 1997 )

mario kart 64 ( 1996 )

panzer dragoon ( 1995 )

daytona usa ( 1994 )

and star trek star fleet bridg simulator ( snes and 32 X version 1996 as well )

Leisure Suit Larry in the Land of the Lounge Lizards - 1987

Batman: The Movie for the C64, Robocop for the ZX Spectrum, Midnight Resistance for the ZX Spectrum, Shadow of the Beast (C64), or Dynamite Dux for the Sega Master System I would say.

There's probably something for the ZX Spectrum that's older that I might have completed, but a lot stuff pre-1985 wasn't "completable" in the strictest sense.

For me it's the same, except with Commodore 64 games. I remember playing through all of the levels for"Neptune's Daughters" (1983) for instance but that game just looped endlessly IIRC.

The oldest titlles I remember finishing are probably a Dutch adventure game called"De steen der wijzen" and Magnetic Scrolls' "The Pawn" (awesome game btw!), both C64, both from 1985.

-edit- correction; apparently, "Steen der wijzen" was published in 1984, not 1985

Super Mario Bros (1985). :) Man i spent so much time playing that game. :D

Zork I, 1980. Though I have beaten the Zork trilogy, so maybe I should get partial credit for clearing 1977's Dungeon...

That's a more interesting question than the one about owning. Hm ... I'm not 100% sure, but atm I can't think of any game I've beaten that was older than King's Quest 1: Quest for the Crown (1984).

Mysteriet på Greveholm (The Mystery at Greveholm) from 1997, a game based on a TV-serie that was shown in Sweden during Christmas 1996.

EDIT: I now remember I've beaten The Flight of the Amazon Queen (1995) as well

Tried Zork once, but I got lost somewhere around that big underground dam.

If I'm not mistaken, it should be the original Wasteland.
I finished it just last year, and I must say it is still far ahaead of several self-proclaimed "modern RPGs" (I'm laughing when typing this, mentally comparing the definition with a few fitting examples I won't mention to avoid the hate XD), especially considering its very open interaction with the environment aspect. That UI, though... let's say that if you can suffer its useless intricacy (under that point of view games have astonishingly improved over time) you will have plenty of fun.

Edit: Wasteland was released in 1988.

Golden Axe (1989) and Duck Tales (1990?)

Loom (1990)

Of the games I currently have "within reach" probably Loom (1990).
At my parent's place there still is my Amiga copy of Dungeon Master (1987).

Ultima III for Apple 2 (1983, but I beat it with my brother and a friend several years later. 1986, I think)

Mortal Kombat 2 is the oldest game I have finished and would be probably the first I have ever managed to beat too. I remember being so psyched about it, good times.