Welcome to LOST! i hope you find all the answers you are looking for here. There are certainly a lot of knowledgable people on here.
Sounds like you are looking for an AT tire. There are a lot of different ones out there. Alot of people like the Bridgestone? Dueler AT Revos. I dont know too much about these, but people love them. General Grabber AT2 is another fav. As is the BFG All-Terrain TA, although these are kind of expensive. I used to have some Dayton Timberline ATs on my Jeep. I liked these tires a lot. They had a nice all-terrain tread pattern, rode nice, no road noise. I had them off road a couple times and was pleased. They are also pretty cheap. I got them for $100/tire including mount/balance. Keep in mind that no AT tire will do very well in mud or really deep snow. Thats just how it is.
As for size, 235/70r16 was the largest 16" dia size that came stock. This is actually a pretty nice size for a stock height Liberty. I would recommend this size, although people also run 245/70r16 usually without any problems. i dont have personal experience with that size on no lift though.
Theres lots of more info on this stuff other people will have. Good luck!

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-Jon-02 Jeep Liberty - LOST Trail Tested Edition - sold
09 VW Tiguan 4motion - current