Billy Hamilton of the Philadelphia Phillies scored 196 runs in 1894.
If you're asking about cricket, and English cricket in particular, the player who scored the most runs in a first class season is Denis Compton.
In 1947 Compton scored 3 816 runs for Middlesex, in 50 innings, with a high score of 246; 18 centuries; and 8 not outs at an average of 90.85. He was a chinaman bowler and played in 78 tests for England.
He also played football for Arsenal, winning the English league in 1948, and the FA cup in 1950.
The second highest number of runs in one season is 3 539 scored by Bill Edrich also for Middlesex, and also in 1947.
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