Iron is the most stable single element (I don't know about combinations of elements, but oh well, not applicable to question). Stars go through cycles where they change the materials of their cores, culminating in iron for the longest lasting and largest stars. The cores of stars naturally have tremedous forces. My question: Has humanity managed to combine to iron nuclei together? and if yes why can we do it but not stars? Alternatively, where would the progression past iron nuclei go, since no element is more stable?
Iron is the most stable nucleus, so in principle all heavier nuclei are unstable and will eventually decay to iron. However if it occurs at all this process is ridiculously slow. For example fusing two iron nuclei would give a tellurium nucleus, but tellurium is (as far as we know) stable to fissioning into iron. The reason tellurium is stable is because there is a kinetic barrier that the tellurium atom would have to overcome to split into two iron nuclei, and outside of the cores of supernovae the barrier is so high that the process doesn't happen on the timescale of the life of the universe.
Anyhow, elements heavier than iron are made in supernovae. If the supernova was an equilibrium process most of the matter in it's core would convert to iron because that's the lowest energy nucleus. However a supernova is not an equilibrium process so it's possible to get elements that don't appear at equilibrium. At the peak of the supernova the immense pressures can fuse the nuclei of heavy elements, and some of these manage to survive until after the peak pressure falls. That's how all the elements heavier than iron got created.
Mankind has (as far as we know) never managed to fuse two iron nuclei. Colliding iron nuclei in accelerators might momentarily produce a tellurium nucleus, but the nucleus would be in an excited state due to the energy of collision and it would immediately fall apart again. Reproducing the conditions in a supernova is beyond us at the moment.