As you can see I have a 2003 F350 Crew Cab, 4x4, Long Bed, Dual Real Wheel truck. I had thought about going up in size on the wheels but decided not to as I have heard issues with going to the 19.5 inch wheels due to limited tire choices. Other sizes are available for the F450 and F550 but they have different lug counts.
I use my truck as a truck. So I don't want to install any adapters which I understand to weaken the assembly.
I plan to install a 2-3 inch leveling kit on the front. This will make the truck look better and allow for additional clearance in the front.

So what is the largest tire than I can squeeze onto the stock 16x6. At least that is the size I believe is on the truck now.

I was thinking I could go with the following:
265/75R16
Diameter : 31.6"
Width : 10.4"
Wheel : 16"
Sidewall : 7.8"
Circum. : 99.2"
Revs/Mile : 639

Or Not as appealing:
235/85R16
Diameter : 31.7"
Width : 9.3"
Wheel : 16"
Sidewall : 7.8"
Circum. : 99.5"
Revs/Mile : 637

 

Without running spacers on the rear, 265/75R16 will rub the sidewalks together. 235/85R16 would be the way to go out of those two choices.
If you want bigger than that, consider getting some aftermarket wheels in a larger diameter.

 

Is there any other sizes that I could consider?

I saw one truck on the Ultimate Picture Thread with 315/75/16's on it with the same wheels that I have. Of course don't know if he had made any other modifications.

 

You might be ok with 255/85R16, you'll be close though. You have to be careful of heat buildup between the rear tires. If you go too wide it generate extra heat retention in the tire, reducing tire life and adding to the chance of blow outs. It you run spacers, you can mitigate the somewhat; you then run other risks. If you go to 17" wheels from a 2007 truck, you get more backspacing and less tire size choices. If you go to a custom wheel, then your choices are much more varied. I remember Stockton wheel used to do some custom dual ly wheels for a reasonable price, that was 20 yrs ago though.

 

Another thing to consider, your stock wheels are 16" X 6.5". Wider tires squeezed onto that narrow of a rim will induce sway as well.