It sort of depends on how you look at it.
Any one of about a dozen Atari 2600 games including Space Invaders, Defender, Combat, Pac-Man, E.T., Cosmic Ark, Vanguard, Baseball, Bowling, Golf and a few others - My Dad bought an Atari 2600 right around the time I was born, and we owned about a dozen games for it before the NES was released. I played these games on occasion, but I didn't really get into gaming until after the NES came out. All of these games were bought when I was too young to truly remember, so they may or may not count.
Mario Bros./Duck Hunt - This is the first game that my family bought when I was at an age to actually remember getting it. This game obviously came with our NES.
Rad Racer - This is the first individual game purchase that I can actually remember, and happened about a month after we got our NES.
Super Metriod - This is the first game I bought with my own money, but I bought it used and only paid $2 for it despite being CIB if you can believe that.
2Xtreme - This was the first NEW game I ever bought with my own money and I purchased it the day after Christmas after getting a PS1 for Christmas. I bought it based on the gameplay from the included demo disc. This game probably isn't held in high regard by most people, but I still have fond memories of it.
So depending on how you want to look at it, any one of those games could be considered my "first game."