Mu Sagittarii is a star system, not a single star. If that can be included, then Eta Carinae should be included, and it has an absolute magnitude of -12.0. It's a star system about 7,500 light-years from Earth.
It looks like the brightest (absolute magnitude) single star visible to the unaided eye is WR 24 (in Carina Nebula). Its absolute magnitude is −11.1 and apparent magnitude is 6.48, so just barely visible.
edit: Rho Cassiopeiae's absolute magnitude is -9.5.