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Handheld it's Pokemon red/blue, no competition.
Edit: although it's actually gold/silver if that counts (released 1999 in Japan, 2000 in North America)
Yes Game Boy was a must in that years, even with just Tetris was great as device and then Pokemon have make the addiction.
Yes, I spent many years with red/blue/yellow, but I have to say, gold/silver were the most bang for the buck. I remember reading so many articles about what the new game would have (day night cycle, radio with daily lottery system, weekly events, new pokeball types, holding/berry systems, etc.) and they delivered on absolutely everything in spades. It was only after I beat the elite four that I realized you could go back to kanto and do 8 more gyms! It was by far the most satisfied I have ever been with a new release to a series. I have since read that the devs were under the impression that it would be the last game in the franchise (lol!) so they poured everything into it. And it shows.