Ideally, I'd like to get pre-assembled boards that just need chips, so that they're ready to just flow the chips in, when they arrive.
That said, I'm interested in kits if that's not an option.
I suspect it would be somewhat of a challenge to solder another chip to a board that already has a lot of other stuff soldered... (that's unless you're looking for manually soldering it, which isn't necessarily any easier but likely has a better probability of success

)
Why would it be difficult? That's literally WHY they make re-flow ovens, and why they call them "Reflow" ovens.
Although I agree that it isn't impossible it would add way too many ifs to the equation. Not all components have the same (reflow) temperature characteristics, some require to be kept at various temperatures for short times while others may need longer times. Then if you have any through-whole components and connectors, especially some plastic materials - they may be affected by that process.
So - yes, it is possible, but you really have to design the whole board for that from the very beginning. And even then you may have to wait until you add all chips and then finish the board and add some of the other components. In other words - you wont be able (at least easily) to really "finish" the board and just wait to solder the last chip.