It's not an open question: Bobby Fischer was the greatest American chess player of all time. He is the only American world champion in history, and won each of the eight US championships that he competed in, closing out the 1963 tournament with a perfect score - one of the only such performances in chess history. He was also a conspiracy theorist, a millenarian and an anti-Semite...
He was also known for the most brilliant sacrifices the game has ever seen. Chess computers and players have since dissected the game and discovered its flaws, but Fischer's 1956 victory over international master Donald Byrne deserves the title "Game of the Century" because it contains the drama and adventurousness that separates good chess from the great. (Fischer sacrifices his queen on move 17, only to chase Byrne's king around the board with a brilliant combination.) Fischer won the game when he was 13.
And, almost since that time, Fischer's life was scrambled by a series of bizarre developments. In the early 1960s he joined the Worldwide Church of God, which gained a reputation for predicting the successive destruction of the world in 1936, 1943 and 1972. (The predictions turned out to be somewhat less than accurate.) And it is not an understatement to say that, after his 1972 victory over Spassky, Fischer went totally nuts. He was supposed to defend the title in 1975, but ended up forfeiting it without making a single move: the World Chess Federation would agree to only 63 of Fischer's astonishing 64 ground rules for the match.
Fischer then disappeared for almost 20 years, reemerging in 1992 to play an exhibition match against Spassky in Yugoslavia, then the subject of UN sanctions that prevented international sporting events. Fischer spat on the order forbidding participation and went ahead with the match, which he won comfortably. The US issued an arrest warrant, and Fischer lived the rest of his life abroad, traveling between Budapest, Japan and Iceland while making a series of increasingly odd statements and claiming to be the "victim of an international Jewish conspiracy".
Commentaries on Fischer; BBC, NYT, NYT The Lead, NYT Chess Blog, The Economist,
Bobby Fischer's Game of the Century
Fischer VS Spassky Documentary
Searching for Bobby Fischer (yes, the entire movie in 13 parts)