A person who plays chess, especially one who is skilled at the game.
An archaic or rare variant formation from 'chess' combined with the agent suffix '-ner' (similar to '-er'), used to denote a person engaged in the activity. This formation is largely obsolete, replaced by 'chess player.'
Most occupations that end in '-ner' are pretty old-fashioned (like 'costumer' or 'gardener'), and 'chessner' is a perfect example of why English preferred simpler words—'chess player' just sounds more natural to our modern ears.
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