What is the biggest tire I can fit on a 2005 F350 without lifting it and what kind of height would it gain?

315- 17 (Hummer H-2) with very minor trimming on the bottoms of the inner fender pieces. They're plastic.

The truck sits on LT 275/70 R18 from the factory.

If you'll put a 2 inch leveling kit on the truck you can put a set of 35" tires under it without any trimming, the 2" leveling kit(oversized coil spring insulators) also doesn't change alignment angles and are cheap. All you'll need to do is have the steering wheel re-centered. I'm just now finishing one for a customer.