No, I don't mean Puerto Rico (BGG, BUY ME!), the number one game on Board Game Geek. I mean Bridge (BGG, WP), also known as Contract Bridge.
I didn't like Bridge at first, much like I didn't like Tichu (BGG, BUY ME!). I can't for sure say why, definitively. Both are partnership games, and there's a lot of strategy behind each one. I haven't really played partnership trick-taking games of any kind beyond Spades (BGG, WP) in high school except for Sheepshead (BGG, WP) and Njet (BGG, OOP). Part of my original dislike of both Bridge and Tichu stemmed from rampant confusion; I simply hadn't had the brain structures in place to grok a game that depended so heavily on good partnership play.
Right now, Bridge kicks Tichu to the curb and laughs at it. I forsee playing Bridge for many years down the line. I also forsee learning Tichu again, now that I've played and am starting to grok Bridge.
I told someone that we learned to play Bridge, and they asked me if it really was an "old person's game." With Bridge, once you've played it for a while, you get to know your partners and their play styles, leading to a more involved game and deeper play enjoyment. Fundamentally, it's not a shallow game. The game of the game is really in the bidding, and a ton of information passes between players, particularly when in conjunction with a bridge convention, which is an agreed-upon method of imparting said information.
What are the other great games? Certainly Chess (BGG, WP) and Go (BGG, WP). Everything else is up for grabs. Some folks might want to include Backgammon (BGG, WP) due to its age, but I've never played. I know I should.
I also read today's Bridge column in the paper, so I know I'm sold on the game.