I'd have to combine a certain balance between enough single-player, enough multi-player and enough sandbox (single-player but sandbox enough to not be too linear either).
Not in order of preference:
1) Mass Effect 3, for the multi-player, and only if it'd be guaranteed that the servers would never be shut down so I could play it basically forever. To be honest I don't think I'd touch the single-player... but it'd be there for whenever I feel like getting depressed. That way, near the end of my life I could play the campaign one last time, get to the end, proceed to be shocked by the unspeakable idiocy I'd just experienced and let the heart attack plus old age combination do the rest. But yeah I'd just play the multi-player until that point.
2) Team Fortress 2, for another multi-player category I also enjoy a lot, but again if it'd be guaranteed that Valve always exists and always supports TF2. Or at least that whatever company following Valve comes in would keep supporting TF2 basically forever. I'd also want to get access to any of the community maps I want.
3) The Elder Scrolls V: Skyrim, for the sandbox portion of my gaming needs, but only if it'd be guaranteed to never crash... and also if I could use any community mods whenever I want. That alone would almost guarantee I'd have 'something to do in that game' until my last breath. But I reiterate, it would have to remain 100% stable and never, ever crash. I know that's technically impossible but I'm giving me the liberty to dream in colors right now.
4) DOOM II: Hell on Earth, to get my FPS fix, and only if I'd also be able to use any community WADs and mods whenever I want. That for me would basically be the pinnacle of FPS bliss.
5) Warhammer 40,000: Dawn of War - Soulstorm, to satisfy my RTS needs both in single-player skirmishes and on-line matches, and only if I'd also be able to use any community mods I desire.
6) Injustice: Gods Among Us, for whenever I feel like playing a fighting game. I'd have to get access to multi-player as well, however. I was hesitant, for a moment I almost went with the original Killer Instinct for this choice, the Arcade version that is. But if I had to play a fighting game for the rest of my life I wouldn't want to stand up playing the actual arcade version for possibly hours non-stop. And honestly Injustice is my favorite fighting game since Killer Instinct anyway, so the choice wasn't tough in the end.
7) Super Mario 3D World, that - for me - would essentially be my perfect platforming game, provided I owned a Wii U. But I am pretending I'd own one for this subject. Although I've actually played the game on my cousin's newly-purchased Wii U and I have to give credits where due, it's my favorite Mario Bros. game in probably a decade, if not the best one I've ever played (if I put my nostalgia aside).
8) Sonic & All-Stars Racing Transformed, obviously, for whenever I crave some fun and lighthearted racing bliss (what a superb game). I admit that I almost picked up one of the Mario Kart games for this choice, but I missed too many versions and 'sequels' of Mario Kart since the SNES version. Not to mention that I never owned a single one since then (since the SNES version). So basically by just renting them occasionally over the years on different Nintendo consoles I haven't played enough of any others in the series to be "sure" if I'd like any of them enough to play them for the rest of my life. I do know, however, that I really love Sonic & All-Stars Racing Transformed ultimately, so the choice was rather easy.
9) Unreal Tournament III, sue me, but I'd pick that one for whenever I'd feel like playing an arena-based shooter. I'd need access to multi-player as well of course (even though I'd primarily play with and/or against bots) but I'd also have to be able to download any of the community mods and maps I want. If that would be the case then yeah definitely UT3 it'd be (almost said Quake III Arena, but ultimately with mods I'd prefer UT3).
10) Mass Effect, the original. I'd need one game in my list for whenever I want to live and re-live a thousand times arguably (barely) the best epic science-fiction journey I've ever been through in video gaming (for me anyway, that game marked me for decades to come). I could easily just be content with the first without ME2 nor ME3 since I always considered the first one isolated from the other two. Oh and I'd also want Bring Down the Sky and access to some of the (rare) higher quality textures mods (it'd be the PC version obviously).
I know myself though, I'm going to re-read this from time to time in the next days/weeks and shake my head as I realize how this or that choice could have easily been replace with another I didn't think of.