By the oldest game I mean in terms of release date not the first game you've actually played. Mine was Super Mario Bros. on the Nes release back in 1985.

All Atari 2600 launch titles. Combat being the best. 

Or Pong, but not the Arcade one.

I really don't know. I know one of the oldest is Crash Bandicoot released in 1996. The same year I was born. There is probably an older game that I've played though.

Pong, I guess.

For me, it has to be either Game&Watch or Pac-man on arcade

Chess, I guess. Hey, you didn't tell me it had to be a VIDEO game. :P

Star Wars: Empire Strikes Back for Atari 2600 (1982)

If you count playing re-releases of games I've played Adventure (1979), Breakout (1978) and... well I think most people have played pong at some point in their lives, though most likely not on the original machine.

I played pong at some tech expo 'history of video games' booth before.

Well if you're going to go in that direction then I've played the royal game of ur.

Rainbow Island for Commodore Amiga.

Since I played Pong back in the days, it's probably my oldest video game. But as many other arcade games (Space Invaders, Asteroids, Pac-Man...) you couldn't really "finish" it.

The oldest game without a proper a story I finished (played all levels) was Soko-Ban.

The oldest game with a story and an ending I played to the finish is King's Quest 1... never finished the Zork-games or most other Infocom text adventures.