Need good free web hosting

The server is actually close to out of HD space at most times, and our bandwidth sucks, so any hosting offer would be on the contingency that you get kicked off if you actually generate any traffic
Now, processing power, on the other hand, we have in abundance...

And there is no way to get around these restrictions and move the web site to somewhere with more HD space and better bandwidth?

To fix the HD space, I'd have to upgrade my server, which would be a pain. As for the bandwidth, there's no need, and I'd rather not switch hosts.