Largest tire I can safely use on my vehicle.?

There is this thing called "scrub patch", which is the distance the center of the tire is off from the ideal or theoretical point at which the tire should be centered. As the over all diameter of the tire increases the scrub patch also increase. To compensate a different off set can be used, but then a problem develops with interference with suspension members.

The jeep is already unstable enough at road speeds and making it more so is not a good idea. Every year I see some body who has done what you are suggesting rolled over or smacked into a tree (some times worse if a person considers driving over a cliff worse).

Well you'll have to check the width of your rims. But looking at the picture, your tires are already filling the well vertically. So if you want wider you'll have to check the width, offset, backspace and then measure how much room there is on the inside so you won't rub.

Go to a Jeep Forum. They'll have your answer 

The largest you can “safely” use on it is the largest size advised by Jeep. 

Anything larger is at your own risk and your insurer will need to know. 

Anything which makes the overall wheel diameter larger is also going to make your speedometer read too low, your acceleration slower, and *might* increase your top speed but will impair the ability of the car to climb hills or tow heavy loads.

If you just want to go wider then you need to understand that will mean the aspect ratio needs to be lower to maintain the same overall wheel diameter. You will also need rims with the correct offset.

265/50 means 265mm wide and 50% aspect ratio. 50% of 265 = 132.5mm. So as long as the wider tyres maintain a sidewall height no greater than 132.5mm they will not have a larger wheel diameter.

If you choose larger rims and wider tyres then you quickly end up driving on rubber which is more like an elastic band than a tyre, with increasingly hard ride (uncomfortable for you, very hard on the suspension and steering components) and far more prone to getting punctured.