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Very: Allegro assai = very fast. Generally read as equivalent to molto, though some writers have taken it in its older sense of “enough, rather”.
Printed after the word it intensifies, almost always in a tempo heading — “Allegro assai”, “Presto assai”, “Adagio assai”, or before it in “Assai vivace” — so its scope is the whole section the heading governs, until the next tempo word.
Assai is the intensifier of eighteenth-century headings — Allegro assai, Presto assai, Adagio assai — and in practice it means what molto means: Mozart heads the finale of his G minor symphony “Allegro assai.” and its first movement “Allegro molto.” in the same spirit. Because some have read it as “rather”, like French assez, keep it to the familiar heading formulas, and add a metronome mark when the exact speed matters.
Mozart, Symphony No. 40 in G minor, K. 550, finale: “Allegro assai.” heads the movement, while the first movement is headed “Allegro molto.” — the two intensifiers side by side in one symphony.
Assai is one of 1760 terms in the Gradus musical dictionary: the words, abbreviations and instrument names that appear in printed scores, plus the vocabulary composers use to talk about them.
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