I don't carry major parts. Most are going to be too big and heavy to be carting around all the time.
I kept my old water OEM pump as a spare. It was noisy but worked so replaced it with a better one and use the OEM to help fill the fresh tank when boondocking and adding water. I carry a spare water heater element.
I like to carry lots of different things to temp patch stuff, like tapes, glues, nuts bolts, and screws and wires. I have a shark bite end cap in case of a pex leak I can temporarily cap it.

We are like Ray, no major parts as weight and storage could become a problem. Greg has a tool box with all the makings for quick repairs as Ray has suggested.

We do often keep the “rest of the package” when you need to replace one piece of something, but the package comes with the whole thing. Such as, the door latch for the screen door. The piece broke that the latch catches into on the door jam. We saved the latch “just in case”. This weekend we were camping with our son and daughter in law and their handle had broken. We had the piece they needed! Earlier the same day, someone had come by looking for plumbers tape. We have it.

Shipping is so quick and easy I carry hand tools and some minor repair supplies but no parts

I carry a pair of small wood blocks that will hold my automatic steps in the down position if the motor fails as it’s done twice now. Other than that, the spares I carry for the truck are a fuel filter, fan belt, idler arm, doser for the DEF system, fuses, and a relatively small but complete tool kit. I also carry a roll of repair tape for the awning, a roll of TPO for roof leaks, a roll of bailing wire, assorted zip ties and a volt meter. Yep, we’re boondockers and frequent travelers to Mexico.

I only carry universal parts like nuts, bolts, screws, electrical connectors, etc. A selection of tapes is there too as is some bailing wire.

Chris, our last long term (11 years) traveling was done on a boat, almost all of it outside North America. And even in those circumstances, I found that aside from 'consumables' (filters, oil, etc.) air shipping and worldwide global connectivity these days made carrying mechanical spares *mostly* a waste of money and space. (The South Pacific was a different story...but I don't think you're going there.) Willie's circumstances are a little different, being a boondocker and Mexico traveler, but for most of us - and maybe you too? - your most likely 'remote' traveling will be within arm's reach of North America's commercial infrastructure. I've begun thinking about a 2022 trip up to Alaska and, even then, I imagine 'short-term repairs' will be the main driver for what I carry with us. (Again, see Willie's list). But my main suggestion for you is to consider 'prep' before 'repair'. Ray's example - a new water pump with the replaced pump still aboard) is an excellent illustration of that. Consider which 'original' systems you consider essential and then upgrade the 'wear' items, with the original items making up your 'spares'. An example of that on our trailer is a sail switch for the heater, as we see cold weather and RV Parts guys tell us that's a weak link. Good luck!

Thanks everyone. Good comments and suggestions all around.