I don't see why not, I enjoy playing games immensely so why wouldn't I during my last 2 months? Last day or two no, but months? Certainly.

Give me 2 months to live from today and I would likely play Dishonored and then replay my favorites, Deus Ex and Morrowind.

If I knew I'd die after the game is released, I'd choose System Shock 3.

Let's see.. well, I wouldn't waste my last days playing a video game, unless I had no choice.
In the instance of the 'death row' type thing, I would find the game that had the absolute most hours possible to log in it and I would play it until I had done everything there was to do.

If it was more like.. say.. I was on my death bed, and I had reached the point where I couldn't do much but play games, I would probably either play various short and sweet games, so I wouldn't die wondering what happened next.. >.> and I'd probably try to spend as much of that time gaming with loved ones.

Contra.

The game music have a lot of memories.

I would spend my last minutes on Earth with my family, there would be plenty of time to play Diablo in hell 8)

Well if I'd die AFTER finishing the game, I would definitely go for unpatched Daggerfall.

WoW. I'll start a new char and do everything solo, every single quest, achievement, and all that stuff. Should extend my life for a dozen or so years.

Daggerfall. And I would do "everything" in that game.

Or Minecraft (same there, everything has to be done, every possible tile (combination) needs to be explored)

Frankly, if I was going to die in the imminent future, I fear both sex and games would be out of my mind at that point. No energy to concentrate on either one.

I'm not one of those guys who want to have sex with the nearest female whenever an asteroid is coming our way, or rush to the nearest coin up arcade/pinball game either. Most probably I'd just pee my pants and be done with it.

Well ok, if I was told to play something at a gunpoint, maybe something casual like Bejeweled or Angry Birds. Or if I was ordered to have sex with someone, Angelina Jolie please. She's a bit old nowadays I guess, but I just need to feel her at least once before I die. I hope she doesn't mind if I peed my pants.

I would probably not be playing a game at all...

Nope, not me. Last time I was about to die, I slapped the reaper and told him "Bitch go make me a sandwich before I beat you again."

As far as games go, I think something epic like Civilization would be tempting.

Fallout New Vegas or a good game of Left4Dead.

Please pm me details of how you intend to pull this off, much appreciated bro

Dunno, not a very original answer but if I didn't have anything better to do with my time and I had to choose one, maybe Planescape Torment, in order to remind me how tormenting immortality can be? And also because it's one of the longer games I've played, granting me more time. ;)

Or maybe one of the mods that I created with a friend or family member, to remind me of the good times.

Or maybe just something pretty trivial but addicting, to keep my mind off the impending doom? Like a casual game or lighter RPG or action game / shooter?

Or preferably something really educational about prison breaks and weapon handling. Or Psychonauts' Meat Circus, to build up enough rage to overpower the wardens, or enough despair to embrace death with a relieved smile. :P

Ghouls'n Ghosts.