I'll do a variation on the question: "My favorite three civs - 1 from each expansion"
Vanilla/DLC
Polynesia. Love the water game.
Wayfinding really sets you free, as exploring without limitations is so fun. Defensive perk is a nice touch.
Maori Warrior is great and gets you to clean up barb camps quicker.
Moai is simply awesome, particularly when you can manage to sustain food/production and rake in culture chains - when that happens there is no stopping you.
G&K
Carthage. Really love the water game.
Phoenician Heritage mountains perk makes playing her so fun, but the free harbors is where it's at and can be a difference maker.
Quinquereme is a very strong vessel that gives you the option to take down a coastal neighbor before they even know what hit them that early in the game.
African Forest Elephant is just great. More expensive? Less movement? Totally worth it for the Feared Elephant bonus, extra strength and you earn GG's faster
BNW
Indonesia. I only just realized how often I play the water game.
Spice Islanders is powerful. Instant happiness, base gold, and trade gold just for settling on another island. Nowhere to settle? Feeling like domination? Allegedly you can also get the unique luxury resource just by taking over someone.
Kris Swordsman give some of the best promotions in the game. Random effect makes it fun and forces you to think your strategies.
Candi allows you to roll in faith, especially later in the game when foreign religions exert more pressure, but also in the beginning with the base faith it provides.
These are three civs that really force a different playstyle and create a unique experience every single time you play each of them. Best of all, the perks they all get are very solid with the only limitation being that they fare better on archipelago, small continents, large islands, etc.