Superlative form of dinky; the smallest, most insignificant, or most cramped version of something.
From dinky (small) + -est superlative suffix. The superlative degree of the informal adjective dinky, which emerged in 20th-century English.
The progression from 'dink' to 'dinky' to 'dinkiest' shows how English builds intensity with informal words—you can criticize something as small, but calling it 'dinkiest' carries a particular tone of affectionate exasperation, especially in British English.
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