three times; thrice
Literal count of three occurrences. Can be used as a noun or adverbially.
He asked the same question three times.
many times; repeatedly; over and over again
Used adverbially to mean 'again and again' or 'many times', not necessarily exactly three. Often found in set phrases like 三度の飯より好き.
Even after warning him many times, he wouldn't listen.
Even after asking repeatedly, there is no reply.
三回 is the standard counter for 'three times' in modern Japanese. 三度 is more literary or emphatic, and also carries the extended meaning of 'many times'.
Standard kanji spelling for the reading みたび.
Mixed kanji-kana spelling, sometimes used for clarity or stylistic reasons.
Compound of 三 (mi, 'three') and 度 (tabi, 'time, occasion'). The reading みたび is a native Japanese reading (kun'yomi).