Religion scholar, James P. Carse, wrote a philosophical tract titled Finite and Infinite Games: A Vision of Life as Play and Possibility (© 1987. ISBN: 978-0345341846). Finite games have a set start and end point. The purpose of a finite game is to win, and since a winner can only be identified by the game ending, rules are created to define the end. An infinite game are bit more alpha-omega; there is no set beginning or end. Play is ongoing; the goal is playing itself. Humans, it seems, love to play and, Carse suggests, need to play.
Jim Macdonald, in a posting titled Fairy Chess (A term coined in World War II Britain where "fairy" meant "whimsical.") describes some playful variations to the game of chess. One version, called Blue Queen, adds a third queen to the board (painted blue so as to distinguish it from the other pieces). Whoever is currently playing can use the blue queen as she wishes. When the other player begins his turn, the blue queen is then his to maneuver. A version called Behemoth adds an indestructible piece to the board, the placement of which (and the destruction of which) is created by random throws of an eight-sided and four-sided dice. Alice Chess takes you through the looking glass: two boards are played with a single chess set.
A version of chess known to many fans of the original Star Trek series is the three-dimensional game played between Mr. Spock and Captain Kirk. There are a number of rule variants out there. Parmen's Page lets you play "Spock's Board" online. A Trek wiki page describes the original variation and its subsequent history in the Trek universe. Diane Duane, author of a number of Star Trek novels, posits a 4D version in My Enemy, My Ally (© 1984, ISBN 0-671-50285-9), where pieces are permitted to travel in and out of play via player-selected time increments on a specially-designed board cube. (D. Joseph Creighton termed this version Hyperchess, which he devised and tested. The rules can be found here.)
Image by Keith Schengili-Roberts
It's not uncommon for archaeologists to unearth games of humans long gone. Senet was an Egyptian game thought to be an earlier version of Backgammon. Written rules have yet to be discovered for it. Senet was so integral to the culture of that time that it would seem its players felt no need to note them. The folks at KingTutShop have provided info on the board, pieces, posited rules, and game variants here.
Hnefatafl (or King's Table) was a precursor to chess and known in Scandinavia before 400 A. D. Played on a marked board with set pieces the goal was to reach a corner square with one's King. The game could play out unevenly, so a game etiquette evolved. Each game played was actually a pair of games in which players switched sides, each player noting the number of pieces he lost or took from his challenger. More info and the game rules can be found here.
The late 1960s saw a movement which its founders called "New Games." The founders sought to challenge traditional game philosophies (text quoted from this site):
Play and physicality were as important to adults as they were to children.
Competition and cooperation should co-exist; but while competition can be important, winning and losing is not.
No one should be left out, eliminated, or unable to play.
Games are living culture, adapted and changed as required.
Play should require no or little equipment.
The rules should be dirt simple and fun.
The guiding philosophy for the New Games movement was: Play Hard. Play Fair. Nobody Hurt. (Note: While out of print, The New Games Book is still available via Amazon as a used book.)
Well, please excuse me. I have to end this post now and go out and play!