maximum numbare of SQL connection strings

If I understood properly, you are asking about number of connection that you are opening using conn.Open(). Well this is know as Conneciton Pooling. Which is Default in On.
whenever conn.Open(); executes, it will create a Connection Pool.
As a good coding practice and standards, you shuld always close the connection. Other wise, after certain period of time you will get error message like Error on maximum number of connections exceeded or connection pool has reached maximum number of connections. You can also configure connection pool limit in your config file as well.
But, as a good practice, always use proper connection open / close.