Monopoly and Scrabble are two favourites, even if it's been a long while since I played them. Monopoly had a sense of progression as well as rewards for those buying certain properties, even punishing other players who landed on "your" spots. It was a great game about financial management rather than flat out getting to the finish line which, while it is a good objective in most games, didn't have the lasting effect that a victory in Monopoly had.
Scrabble on the other hand can be fun no matter how much vocabulary you have. Playing against someone with great intellect can be a match of wits, whereas bending the rules and having a laugh can still be just as fun. It's a game that's rules are flexible if not being played seriously, and doesn't feel overly competitive even if you do take it seriously.