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Shelomo HaMelekh (King Solomon) famously wrote in Mishle'i 24:16 (Proverbs):
כִּי שֶׁבַע יִפּוֹל צַדִּיק וָקָם וּרְשָׁעִים יִכָּשְׁלוּ בְרָעָה
For a righteous man can fall seven times and rise, but the wicked shall stumble upon evil.
I've looked through Miqra'ot Gedolot, Perush HaGR"A, Perush Rabbe'inu Yonah and Perush HaMaLBI"M and have not seen anyone ask/answer: why seven and not another number?
As noted by @DannySchoemann, Ralbag writes in his commentary to Proverbs (24:16) that in this context, seven simply means many:
הצדיק יפול פעמים רבות ויקום בכל פעם ופעם
The righteous person will fall down many times, and get up every time.
This seems to be the implication of Ibn Ezra (there) as well.
This is also stated by R. Sa'adya Gaon in his commentary to Proverbs there (R. Qafih translation p. 187-8). This is also stated by Rabbenu Netanel Beirav Fayyumi in Bustan al-Ukul (R. Qafih's translation pg. 41.)
In regards to the question:
Why not any other number greater than one (or greater than, say, five if you want to get the point across that it's "a lot")?
The answer seems to be that in the Torah, the number 7 is used to connote 'many', cf. Rav Sa'adya Gaon's translation of Genesis (4:24) (Arabic).
The question then becomes why in Torah (and the rest of Tanakh) the number 7 is used to mean 'a lot'.
The answer to that is presumably what R. Moshe Shamah writes:
In the ancient Near East, dating back to centuries before the Torah, the number seven was considered most prominent, representative of completion and perfection. (Recalling the Covenant p. 1062).
In kabbalistic tradition there is an ascent of seven levels, from the sfirah of malchut up to the sfirah of chesed. The idea behind the fall is the attempt on their part to attain the next level. A fall in this context isn't literally a fall, but a realization that they have to rise above the current level. It is called a fall because of mankind's fall due to Adam's sin regarding the fruit of the tree of knowledge.
The simplest answer is "seven" is a generic biblical term for "many." See the curses at the end of Leviticus -- "I will give them seven more punishments." Some commentaries try to count out exactly how it makes seven; others simply say it just means "a bunch."
