What is the largest tires that can be put on the standard 16 inch rims and still have decent turning radius and no rubbing of the tires in the rear?

 

Well I'm not sure what will actually clear in the back, where you typically run into issues plus-sizing is near full-lock on the front tires....

The factory rims are pretty wide, I know there's a 16x7 and 16x8 factory wheel, mine are the 16x8's, which can hold quite a bit of tire...you're going to want to keep the overall diameter pretty close though for clearance and to keep the speedometer close to accurate.

I would imagine the car could clear a 245/50R16, which is two steps wider and one to two steps lower in profile. Also that's a popular enough size to give you some options.... The next widest step that has fair availability is a 275/50R16, which is a LOT wider and taller and would most certainly rub, at least on the front if not all the way around.