There are many words and phrases that can be used as synonyms for the expression "not give a tinker damn". Some of these include not caring at all, not being bothered, not giving a fig, not being interested, and not giving a hoot. Other similar phrases might include not giving a damn, not giving a rat's ass, or not giving two cents. Ultimately, all of these expressions convey a sense of complete indifference or disregard for something or someone, and can be used in a variety of different contexts to express this sense of apathy or lack of concern.