The longest time owning a game without beating it? I would have to say that would be Twisted Metal Black for PS2. I had that game for over 10 years, I got it as one of the first games for the PS2 when I was 12. Twisted Metal games are among the hardest games to beat because the computer is a cheating bastard. For those that don't know, the twisted metal series are a group of games were you use weaponized cars to compete in a tournament to the death. You have to destroy all the other cars in the tournament on ten stages, and fight the occasional boss. The enemy AI is cheap and broken, all the enemy drivers come after you, the human player, and never attack each other, even though it's supposed to be a every man for himself death match. And the enemies have unlimited missiles and other weapons, while the player has to find power-ups lying around. The game takes great coordination and precision to win, and I used to have terrible coordination. So I sucked at the game.
Recently, I decided to pick up and play the game to see if it works to blow off some steam in finals week. It did, and much to my surprise, my coordination has greatly improved, I'm still not great,but now I'm actually winning matches. I can even execute some of the energy weapon combinations like Freeze, Shield,Mine, Invisibility, and Hop with ease. Now I've completed some of the story mode with a few of the characters.
My goal isn't to beat Twisted Metal Black, my attention is on more recent games and not retro, but I did read they are releasing a new Twisted Metal for the PS3 this year, and I will be one of the first to pick it up when it is reduced in price.