After death which of our body's metabolic processes will continue functioning for the longest?
My answer isn't researched; it is speculative.
I interpret metabolic processes as anabolic or catabolic processes. I assert that only anaerobic processes can continue for a long time after vegetative death (the epithelia could house a counter-example of a post-mortem process, but I can't think of one), and I speculate that only catabolic processes are likely to have sufficient pools of reactants to continue after death. So we're looking for a catabolic process with a large reactant pool. Therefore I think that the action of digestive proteases will continue for the longest time, because these enzymes are very stable and could have a lot of substrate around (including, at a very slow rate, cleaving each other.) The neutralization of stomach pH would inactivate this process...I don't know how long that takes.
If something like blood clotting is considered a metabolic process, then probably that continues the longest. The binding of CO₂ by hemaglobin is probably faster than blood clotting. Again, I don't really consider those processes to be metabolic.