Understandably, there must have been many! Any information on any example would be much appreciated.
Not to discourage other answers, but I'd recommend you refer to the Calendars section of the FAQ which covers the topic pretty thoroughly. Your particular question seems to be covered here.
Well.. Are we talking strictly in the Judeo-Christian world/Anglosphere or more broadly? There are (potentially) as many calendars as there are ethnic and/or religious groups.