I can only speak about my favorite genres:
Action games: this one is easy—Devil May Cry 1.
Shmups: Hmm… Battle Garegga? I'd actually rather go for a CAVE shmup seeing how influential they were, but I could be biased: DoDonPachi.
Fighting games: hmm… a bunch could go in here, but if I have to choose one, it should be Street Fighter II. Biased pick is Guilty Gear Plus R.