Tire height is directly related to tire width. You're getting into the Sidewall Aspect Ratio here.
For example: The G6 came with 2 size tires for the 17" Rims. 215/55R17 and 225/50R17
My car came with the 215/55R17, It was 215mm wide with a sidewall ratio of 55% to fit a 17 inch rim. So the sidewall from rim to tread is 55% of the width of the tire. If we convert 215 to inches we get 8.46 inches, 55% of that is a tire height of 4.65 inches
The wider tire is 225/50R17, so being 225mm wide we get 8.85 inches, 50% of that is a tire height of 4.43 inches. So the wider tire is only about 1/4 inch shorter.
But if you were to get a 225/55R17 then the sidewall would be 4.87 inches tall. But you can't find a performance summer tire in this size easily.
With that being said, you can fit as wide a tire as the rim would take, and the wheel well would clear, but would need a higher sidewall ratio to make up the sidewall, Keep in mind that you can get into some oddly sized tires and start spending more money.
I would recommend checking out TireRack.com You can search by car, or by tire size and get a good idea for tires, sizes, performance ratings. It's a great place for information and to really make an informed decision on tires.
I hope this helps!