It's going to depend on whether you stick with the factory rims or not. If you do, you could easily go up to as tall as 33.3-33.5", and potentially as wide as 13.5" or 325. But the devil will be in the details, in terms of your specific truck, and the offset of your rims, and the specific profile on the tire model you choose as some are wider at the shoulders than others in the same size, and height varies by a few tenths of an inch in any given tire size. You might even get 34" tires to work, but it would depend on what tire.
You may find both of those a little short. The 285/60 will be 31.5" tall, and the 285/65 will be about 32.5. I have a bone stock suspension on my 2011 Pro4x and the OEM tire was a 275/70R18, which is 33.2" tall. A leveled truck is easily good for 33.5" height, and something on the order of 11-12" of width, generally speaking.
Also of consideration is the year and configuration of your truck, as early trucks can be "leveled" more in the front (they require more lift to be level) before developing coil bucket contact. Later models can only be leveled 1.5", and even then some trucks get CBC and need aftermarket Upper Control Arms to clear the coil buckets.
Gonna look like little tires on a big truck. 285/60/18 is a 31.4" x 11.2" tire. My 275/70/18 (stock Pro4x tires) are 33.1" x 11". The 305/65/18 is 33.6" x 12" which would look good. Plenty of guys run 325/60/18's on leveled trucks.
I would not go higher than 275/70 personally. I have 275/65r 18 and they would rub. Level kit trucks do not have very much more clearance unless fender wells are modified. As for frame rub Rough Country has wheel spacers for GM 6 lug that just so happen to fit perfectly on Titans.
Keep in mind going big may also result in an increase in speeding tickets. Speedometer will say 65 and you will be going 67-72 depending on how much larger your tires are. If your going all terrain keep in mind there is added tread depth which may add another 1/2 inch circumference.
I run the Nitto Terra Grappler G2 on my rig. Only drawback to those tires are they like to collect rocks in the tread.
And if it is just a leveling kit and stock wheels bigger than pro4x stock will rub without further modification. My suggestion is not inaccurate. Test fit the rims and tires just to be sure. But likely shops will turn you down or make you sign a waiver to install bigger tires.
With a level, guys run 325/60R18s. Have to pull the mud flaps, but some guys do it without any cutting. And 305/65s will clear with a level. Even without a level, I could go to 305/65 and only pull my mud flaps.
+1 to rwhite188. Picture shows filled up well that looks good with no sign of rubbing. I'm not slamming anyone here I am just providing a suggestion based on the original topic. My suggestion is geared toward the possibility that perhapse a reader does not want to tear cosmetics from their beloved vehicle. I did not say anything was a bad idea in general, just "Requires Further Modification." What kind of tires do you all suggest? Toyo is a pretty good tire. Toyo owns Nitto. Michelin owns BfGoodrich. If price is an issue check into Cooper Discover RSX. Anyone have an opinion on Mickey Thompson or Dik Cepek? General is owned by Continental. I have never been a fan of General but have heard amazing stories of how the grabbers last for the class of tire they are. And yes wheel spacers are a bad idea. They will over stress the hub assembly and will prematurely result in unintended negative camber. My boss runs them just fine but he has the entire RC 2.1 lineup on his lifting it 6 inches with 1' wheel spacers on 35/12.50 20s.