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Mine was this Grandstand self contained game "Scramble" in the 80's. I couldn't tell you how long I spent on it as a child but it was probably quite a lot. It was very minimal but at the time I thought it was amazing :)
http://img229.imageshack.us/img229/2421/scrambleyk3.jpg
After that I got a Nintendo Donkey Kong Game and Watch which was equally absorbing. I still have them somewhere.
My first machine that could play multiple games was an Acorn Electron computer which I needed for "educational" purposes. Needless to say, it got used for games and the occasional typing up of programs from Electron User magazine which more ofter than not, didn't work.
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My memory only goes as far as Double Dragon :3
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My earliest memory of playing a game was taking turns playing Donkey Kong Jr. with my mother on the 2600 when I was about 3/4 (it was 1990). She and my father had the thing already for a few years and would play it with friends (big projection TVs, old pinball machines, jukebox, 2600 fit right in).
The first that was all mine was the Apple IIc I got a year or two later.
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This first for me was a nes and a ton of games that my grandmother just gave to me.
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My first console was the NES with Mario bros 1/duck hunt cartridge.
