No stock rims, or aftermarket with the same ~5.5" backspacing, allows for the biggest tires.
With a level kit and stock rims, some Titans rub a little with 315/70/17s but some don't. So mud flap removal, and in some cases the pinch weld mod. I have a
poll going on this, but I don't think all that many people have tried this size, so the poll probably won't give a definitive answer. I think it is tied 1:1 rub vs. not rub at the moment.
No lift, stock wheels, and no mods BFG AT KO LT285/70/17, 32.8" diameter and
I think the Nitto Terra Grappler 295/70/17. 33.19" diameter.
With level kit, stock wheels, and no mods, Toyo MT LT295/70/17 33.5" diameter. Nitto Terra Grappler LT305/70/17 34" diameter (maybe).
With level kit, stock wheels and pinch weld mod, most 35 x 12.5 x 17
Disclaimer: Although some people use tires on rims of smaller widths than manufactures' recommendations, it is not something I am recommending to anyone.