How can I prevent or mitigate the front wheel coming off the ground when climbing steep hills? On one of my bicycles (a 2012 Trek Madone 4.7), when climbing very steep hills, I have had problems with the front wheel coming off the ground. This creates an extremely dangerous situation because when this happens one loses all steering control.

Do I have any recourse beyond choosing a less steep route, getting less fat (catch 22), or using a different bicycle?
Move your weight further forward to keep the front wheel weighted. Shuffle forwards on your seat and bring your chest closer to the bars. Standing can help for the steepest parts, but can cause your rear wheel to slip on loose surfaces.

The front wheel is lifting as when your bike is on a slope the wheelbase is effectively shortened, bringing your weight closer to the rear axle, and the most vertical part of the wheelie is already done.