You can go wider than two sizes on a factory rim. I used to run extreme examples like a 12.5" wide tire on a 7.5" rim, though this was for off-roading on purpose built jeeps and more for rim protection. However, your biggest concern is going to be that you will crowning the tire, meaning, the tire will be squished like a donut and the footprint will be decreased by running on the center of the tread rather than the whole entire tread section. This will cause accelerated wear in the center of the tire. Handling may be decreased as your effective contact patch will be decreased. Another setback is that you now have a bulging sidewall and the factory inset or offset of the wheel will not bring this extra sidewall bulge out away from the inner fender and suspension components. Rubbing at full or close to full wheel lock may happen or when the suspension cycles up and down... Mostly on compression when the suspension gains negative camber. Another negative is that the overall diameter of the tire increases and thus throwing off the calibrated speedometer reading.
A few sizes wider may not increase this effect very much and I would say keep it at a few sizes larger and no more.
Another way is to not only play with the width but also the height or aspect ratio of be sidewall. This can usually get a more aggressive appearing tire without much negative drawbacks.
Figure out your wheel width and then go to the tirerack.com. Figure out what tire you like and then go to the specs of that tire and look at the section widths and heights with your measured rim width.
Hopefully I'm not confusing you. Lol