I fluctuated between 300 and 315 lbs when I began riding after a 20 year drought. Not only was I concerned about finding a bike that would carry me, I wanted to load 50-60 lbs of gear on it and go touring.
I put together a Fuji Touring bike frame with parts I thought were suitable to the task.
It has been rock solid reliable. I've never broken a spoke or had any other problems. I was especially concerned about the wheels. I didn't want to be worrying about hitting a bump and potato chipping a wheel while rolling down a hill at 35+ mph. I have no fear with the 48-spoke wheels I use on that bike.
Here's an accompanying article I wrote about the various components I used to build it (click the link):
How to haul a big man and his gear on two wheels
I ride every day and I'm now down to 250 lbs. I treated myself to a new Cannondale Synapse for a more sporty bike for local riding. I still tour on my Fuji.