As fast as possible; with great speed; a dialectal or archaic phrase combining 'as' and 'fast'.
From Old English 'as' (conjunction meaning 'to the degree that') + 'fast' (from Proto-Germanic *festuz meaning firm, solid), creating a superlative speed phrase.
Asfast is almost entirely obsolete in modern English, but you might spot it in very old literature—it's the kind of word that shows how older English liked to smoosh words together for emphasis.
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