I like plenty of games that don’t use dice, but none of them are role-playing games.
It gets trickier if the question isn’t using “system” in the sense of “rules system” but rather in the sense of a mechanic (though since yesterday’s question was about dice mechanics, maybe it’s not… who knows? Crunchy stuff like systems and mechanics are things I dig deeply into; I’m more of a fluff guy). I like Hero Point systems (whether they’re called Bennies, Hero Points, Inspiration, etc.) because I enjoy mechanics that let players do cool things that may not otherwise be allowed by the rules. Anything that allows the players to think “Wow, that was awesome!” is something that helps make the game session better for everyone.