In the game you take on the role of a fantasy character in a fantasy setting, so if that counts, King's Quest VI.
If it was the first one a console, then Zelda:LttP or Earthbound.
If it was the first one to really show me how deep, and involving the story of a game could get, how impactful the music could be, and made me really feel the struggle of the characters, Xenogears.
I think all those games had a strong influence on my taste in games.