Personally I think the "must haves" list is pretty small, but everyone has different needs. For me the reason I built the trailer in the first place was I needed space for gear since my wife and 4 kids are typically with me. We just ran out of room inside the truck. But the trailer also allow room for other items that are important for camping and are on the short "must have" list. Things like water and fuel. The only other must haves in my opinion are a comfortable place to sleep which can be a good ground tent and sleeping pad, but for us is a RTT. One other thing that I feel is a necessity is a solid way to stabilize the rear of the trailer if you are sleeping in a RTT on the trailer. A Multi-axis coupler is a must have I think. Lastly is a trailer that is capable of going anywhere I can take the truck. I won't necessarily take it *everywhere* I can take the truck but I wanted it capable of doing just that. I built the trailer specifically to be able to take it through the Rubicon. I feel 100% confident that it can do that, although I haven't been on the 'Con since the trailer was completed.
I just use lots of bins to keep gear dry and protected. It's simple and works.
I don't have a fancy kitchen or galley. I have a 20 year old Coleman white gas stove. Works great and is easy to pack.
I carry two 5 gal water cans, one 5 gal fuel can
Full size spare for the trailer, all trailer tires are the right size for the truck too if I need to cannibalize tires from the trailer in a pinch.
Portable CO2 for filling tires and running air tools.
I have a fridge and while I feel it is a luxury, not a requirement, I don't know what I'd do without it. I love my fridge, but it is carried in the truck. If you don't have room in the truck then have a spot on the trailer.
I don't feel like electrical power is a requirement. It's a luxury.
In the end, I believe the goal should be to carry the gear you need to places that the average person isn't able to get to. Afterall, exploration is about seeing new places.