What orbit is the smallest or takes the least time to complete? What orbit in our solar system would take the least time to perform 1 orbit?

What is the fastest possible orbit to complete in the universe?
In general, the closer you are to the primary, the shorter the orbital period, except of course that you can't orbit inside the primary. So, taking the primary to be a sphere of radius $R$ and density $\rho$ we find that the mass is $$\frac{4}{3}\pi R^3\rho$$ so that the acceleration due to gravity at the surface is $$G\frac{4}{3}\pi R\rho$$ For a surface-grazing orbit, this has to equal $R\omega^2$ where $\omega$ is the angular velocity, so we get $$\omega^2 = \frac{4}{3}\pi G\rho$$ Now the orbital period $P$ is $2\pi/\omega$ so we get  $$P = \sqrt{\frac{3\pi}{G\rho}}$$
So the answer, at least for spherical primaries) is going to be a close orbit around the densest object in the solar system. The densest planet is Earth! 
There are denser objects, such as the LAGEOS satellites, but I'm not sure if it is possible to orbit LAGEOS under its own gravity, because of the perturbing effects of Earth's gravity.