I'd recommend a touring or mountain-bike with 26" wheels. You can run big tyres 26x1.95" to 2.5" on them and not have to worry about potholes and trashing the wheels. Then run them at 50-60psi and you won't be compressing the tyres much at all.

FWIW, I started 1-jan-05 weighing in at 245lbs and hopped on my carbon Specialized Allez Epic and rode on my spare race-wheels without any problems. I did have to bump up the tyre-size to 700x26/28c and run 120/130psi to hold my weight, but that was the only modification I needed to make. Although I did end up bending a couple rear axles on the wheel from hopping kerbs and speed-bumps...