I recommend staying under a 29.5 inch tire MAXIMUM if you want to stay stock. Any bigger than that you will have to re-gear and start cutting into things, the problems I ran into after buying my 2011 KK with 31" tires are: tire rubbing against the metal frame (body) of the car, and significantly heavier driving experience.
Having 31" meats on a stock KK, pretty much kills it's capability. ALSO, you have to keep in mind that running bigger tires without re-gearing will eventually wear out your transmission prematurely.
I am in the midst of downsizing my tires back to stock 29" 235/70/16 - It's a little bit of a challenge to find mud tires this size but its worth having a fully capable jeep that won't wear out prematurely.
Keep on Jeepin!