Once, "twice," and "thrice" are the three only words of their type, and no further terms in the series have ever existed.
It is really uncommon to use the word “thrice”, but it is somewhat common to see in Old English texts. It is more common to say “three times” than “thrice.” It is very common, however, to say “once” or “twice.” It is also common to say “one time” or “two times,” though. Twice is more often used than two times when used in a sentence.
Some people don’t count “thrice” as a word, but it is completely grammatically correct and usable, just uncommon. Other words we use to express numbers without naming the numbers are common, too. Half a dozen refers to the number 6, while a dozen refers to 12. The words nought, nil, zero, o, and love refer to the number 0 and a score means 20.
Some of these terms, especially a score, are rarely used. It is fun to try out new vocabulary and impress your parents who think all you do is play video games and use up their money.