what is the largest tires i can put on my truck

i want to put some bigger tires on my  chevy silverado 1500 2009 i was looking at some 33" on 17" rims would they fit the wheel well or would they be too big for my truck without lifting it i currently got stock 265x70x17  

tight fit, you might wanna put 3" suspension lift on that  
________________________________________
Vehicle- 2001 Dodge Dakota Quad cab
Color- Black
Engine- 4.7L 287CI Magnum V8
Tires- 265/75R16 BFG AT T/A KO

It's usually the front that causes the problems, turning may cause the tires to hit suspension components.

As for the biggest tire, that really depends on how much you want to spend.  Really big tires demand new gears to keep performance close to what you have.  Your stock tire is about 31" tall so a 33 is not a major change.  You may have to do some fender trimming unless you're not taking it off-road.

If you decide to go bigger then 33s and no lift, the get the saw out and start trimming and get the wallet out for new gears.  Go bigger then 35s and have a big, heavy right foot, you will want new axles to match.  

if you are looking for a little extra room without spending to much on a lift you could always get a leveling kit. i did one on my truck for a little extra height and tire clearence. its really simple i put 2 in blocks in the rear axle and cranked up the torsion bars. ive heard of guys getting as much as 3 in of lift by cranking the torsion bars. i would not recomend 3 in because u get and extremly rough ride but ive had great results with 2 in. you can pick up a kit on ebay with new keys and blocks for around 150$ but can go cheaper if you dont get new keys and buy the two inch blocks at autozone which is what i did. ik that you have a way nwer truck then mine but you can find leveling kits for them and i also believe that those trucks still have torsion bars so this should work.  
Chevy Trucks!!!
Got Sand?
Silver Lake Sand Dunes

Your truck has got strut suspension up front. Many companies build leveling kits. I would recommend either a Tuff Country 2" leveling kit or a Rancho Quick strut. The quick strut gives you a whole new strut assembly while giving you 2" of lift.You could also do this kit at home. Just make sure you have an alignment done when you're done. As long as you stick with a wheel that gives you a high (positive) offset, a 33" tire will fit with minor trimming. I do this work for a living, hope this info is helpful.