As far as international, NBA, and US College Basketball is concerned, there is no limit as to the number of overtimes that can be played.
There are 733 basketball leagues excluding the NBA
There is the NBA and there is a league with players not quite at the NBA level. The NBA is the major league with the professional basketball players. There are also college leagues, and elementary schools even have leagues.
It was invented on August 3, 1949. The two basketball leagues back then merged together. The two basketball leagues were the Basketball Association of America (BAA) and the National Basketball League (NBL).
Basketball and baseball
Yes, there are basketball leagues for women outside of the United States.
Youth basketball leagues are often run by your local park service or community college. Contact them to ask about the schedules and sign up requirements.
the ABA for American Basketball Association
There are too many to count. There are dozens and dozens of basketball leagues, a few American football leagues, soccer leagues, many baseball leagues, and so on. There is no kept count of professional athletes.
Yes, Raleigh does in fact do have a youth basketball league available. Not every community have sports leagues but there are several other communities, including Bellevue and Albuquerque.