How can I fit 6 bikes on a hitch rack?
Here's a rack that will do six. I have no personal experience with this one and the fact that neither of the top two rack companies (Thule and Yakima) do a six bike rack makes me think you should approach with caution. 

[Link removed due to the old link being expired/stale. The link was to a "Northshore 6 Bike Rack"]

I have seen, with mixed success, 4-bike bike racks where folks have strapped an additional bike into the mix. Again, proceed with caution.