Perhaps a fast spinning star (not spherical) where a black hole forms in the center, but the event horizon is smaller than the equatorial radius but larger than the polar radius?
A ring of particles, e.g., Saturn's rings, can form around a massive center, and can theoretically even exist without a massive center to orbit around; but such a ring is unstable on a time scale that increases with its mass density (that's part of how planets form from dust rings around their parent stars). I strongly suspect that a toroidal star massive enough to generate energy by fusion would be so unstable that it would last days at most -- but doing the calculations is beyond my skill set! Maybe the torus would be stable if it had an enormously strong current running through it. Sounds like a good SciFi gimmick.