Recently I've really gotten into chess variants, especially 4 player chess. I've played over 2600 4 player chess games and over 1500 chess variant games. In January 2022 I was ranked #4 in the world by rating (3001) in the most common 4 player chess variant (FFA Rapid). My rating is above 2000 in Horde, Giveaway, Atomic, Crazyhouse, 3 Check, King of the Hill, and Chess960. I'll give some comments and resources on chess variants below.
The variants I've played on chess.com/variants are:
4 player chess (Wikipedia; chess.com): I participated in the 2021 Online World 4 Player Chess Solo Championship, where I scored 7th out of 16 players. The three standard modes of play are Solo (only 1st place wins and 2nd, 3rd, and 4th lose), Free-For-All (same rules as solo but 1st and 2nd both win and 3rd and 4th both lose) and Teams (the players on the opposite sides of each other are a team and must checkmate one of the members of the orthogonal team to win). The 4 player chess server on chess.com also has dozens of unique and fun variants, and new ones are being constantly created. 4 player chess is my favorite variant thus far, and it's the most popular variant on chess.com
Horde (Wikipedia; chess.com): White has 36 pawns and black has the standard chess set. One of my favorite variants, and the first one besides 4 player chess where I reached a rating of 2000.
Giveaway (Wikipedia): The objective of each player is to lose all of their pieces or be stalemated. This is a very weakly solved variant where 1.e3 is winning for white, but in practice this move is simply a good first move and doesn't guarantee winning. Many first moves for white and black lose by force.
Atomic (Wikipedia): All captures result in an "explosion" through which all surrounding white and black pieces other than pawns are removed from play, as well as the capturing piece and the captured piece. The following links discuss atomic openings: 1, 2
Fog of War (Wikipedia): A player does not see the entire board – only their own pieces and the squares that they can legally move to.
3 Check (chess.com): Checking your opponent 3 times wins; otherwise, all the rules are the same as in regular chess.
King of the Hill (chess.com): Bringing your king to one of the four central squares of the board (d4, e4, d5, e5) wins; otherwise, all the rules are the same as in regular chess.
Blindfold (Wikipedia): Regular chess rules, but without being able to physically see the board or pieces. Moves are communicated between players verbally or written down.
Chess960 (Wikipedia): Regular chess rules, but the starting position of the pieces on the 1st and 8th rows is randomized.
Bughouse (Wikipedia): Played on two chessboards by four players in teams of two. Normal chess rules apply, except that captured pieces on one board are passed on to the teammate on the other board, who then has the option of putting these pieces on their board.
Chaturaji (Wikipedia) is an ancient Indian chess variant with 4 players and two dice throws on each turn. The goal is to collect the most points, like in contemporary 4-player chess, but the king is treated like any other piece, so there are no checks or checkmates.
Chaturanga (Wikipedia) is an ancient Indian chess variant, thought to be the common ancestor of chess, xiangqi, shogi, sittuyin, and makruk.
Courier chess (Wikipedia) was a popular chess variant in Medieval Europe.
Gothic chess (chess.com) is a variant of Capablanca chess which, like Capablanca chess, incorporates two additional pieces: the archbishop (bishop and knight combined) and the chancellor (rook and knight combined). It is played on a 10×8 board. From what I've read, Gothic chess and a few other variants of Capablanca chess improve on the original by changing the initial setup of the pieces so that none of the pawns are undefended at the beginning of the game.
XXL Chess seems like a derivative of Chess on a really big board (Wikipedia).
Racing kings: The white and black kings race to get to the 8th rank. Analogous to playing Mario Kart on a chess board.
The list of chess variants on Wikipedia is amazing! Notable and funny variants:
Chess boxing: Contestants compete in alternating rounds of chess and boxing.
Beirut chess (variant of atomic chess): "The game is played using the standard chess pieces and board, with each side having secretly equipped one of their men with a "bomb"—which can be "detonated" at any time, wiping out all men on surrounding squares along with the bomb carrier".