Who operates and uses the largest supercomputer in the US?

Answer 1

Currently (June 2010) the fastest supercomputer in the US (and, indeed, the world) is Jaguar at Oak Ridge National Laboratory. China's Nebulae at the National Supercomputing Center in Shenzhen has a higher theoretical peak performance, but loses out (by a small amount) in real-world tests.

If you mean physically... I'm not aware of any listing. Many distributed computing solutions can grow quite large (though physical size does tend to impact performance to some extent for most problems). Virtual supercomputers such as the Great internet Mersenne Prime Search project or SETI@Home extend literally worldwide.