I would be willing to bet that there's not a single county in the US with a population density higher than the population density of its largest city. (obviously city-county mergers like NYC, Indianapolis, Jacksonville, etc would have a ratio of 1.) Indeed, most counties that have any town of > 20k people and that are not dominated by a huge city probably fit your criteria.
I'm having trouble finding data on this where all the populations are from the same time, but Forsyth and Gwinnett counties in Georgia might be counterexamples. (Their largest cities, Cumming and Peachtree Corners respectively, are hardly cities so much as just bits carved out of North Atlanta suburban sprawl.)
