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That depends on how much cutting you want to do . I think that 35x12.50 tires look perfect on a 4" lift. That's just my opinion though. If you want to go bigger than that, there is a good chance that you will need to start trimming the fenders so the tires don't rub.
It really depends, but if you only intend on using it as a mall crawler, then you could fit some 39s under there, but when you hit a speed bump, they'd probably hit the fender, but for actually off road use, 36s would be the biggest I'd go with a 4" lift on a 1/2 ton truck.
36s would fit without any problems, and any bigger than that you will need to upgrade axles. You will need to regear as it is for 35s or bigger if you want any kind of acceleration power or mileage.
I just put on goodyear 36" hummer tires on my truck with a rancho 3". The lift actually measured 3.75" and they don't rub. However, I don't do extreme flex either.