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Old English ænig "any, anyone," literally "one-y," from Proto-Germanic *ainagas (cf. Old Saxon enig, Old Norse einigr, Old Frisian enich, Dutch enig, German einig), from PIE *oi-no- "one, unique" (see one). The -y may have diminutive force here.
Emphatic form any old ______ (British variant: any bloody ______) is recorded from 1896. At any rate is recorded from 1847. Among the large family of compounds beginning with any-, anykyn "any kind" (c.1300) did not survive, and Anywhen (1831) is rarely used, but OED calls it "common in Southern [British] dialects."