How many Super Bowls have the Cincinnati Bengals won?

None.

The Bengals have played in two Super Bowls and lost both. In Super Bowl XVI they lost to the San Francisco 49ers 26-21 and in Super Bowl XXIII they again lost to the 49ers 20-16.
The Cincinnati Bengals are 0-and-2 in Super Bowl appearances, having lost twice to the San Francisco 49ers.