The most common amount of days in a week is seven originating from the planetary week where the weeks are named after the seven known (at the time) planets. This also conveniently aligns with the biblical week. What however is not that universal is the first day of the week. Over the different cultures using the seven day week the first day ranges between Friday and Monday.
Despite the differences it doesn't make the standard week too interesting (btw. According to the ISO standard the week starts on Monday). There are many other length of days used for the week as well. But probably the most interesting week system is the one used in the Balinese Pawukon calendar.
Pawukon is a time period of 210 days that is simultaneously divided into weeks with 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 days. If that isn't confusing enough there are also special rules for the weeks with 4, 8 and 9 days as 210 is not dividable by them. To make things work out additional days need to be added to those weeks: the second to last day of the weeks of 4 and 8 are repeated on the weeks that contain the 72nd day of the Pawukon while the first day of the 9 day week is repeated three times during the beginning of the Pawukon. Furthermore, only the 3, 5, 6 and 7 day weeks are repeated cyclically. The 1, 2 and 10 day weeks start according to some elaborate rules based on the ritual days and their conjunction with other weeks' days.
I will never again complain about datetime math as long as I don't ever need to implement a Pawukan calendar.