Let's start with a fairly obvious fact: human computers are produced every day by the process of birth. I'm stating this just to emphasize that this question is really a question of "are human made artifacts different, in some fundamental way, from humans themselves, which are themselves produced by a process".
For a materialist, the answer is an obvious no; and thus there are no obstacles in principle to making strong AI.
For a dualist, the situation is open: if there is a mechanism by which we can induce a soul to inhabit a human constructed entity, then we could have strong AI. Only if we have souls, and there is something that prevents us from setting up the conditions whereby a soul would end up inhabiting a human made entity, would there be a fundamental limit.
Usually, we think of strong AI as embodied in a computer. What if we were able to construct artificial life-forms biologically? We can currently construct whole viruses, move nuclei from cell to cell, make hybrids etc. To me, it's harder to argue that this type of artificial biological system would have a fundamental difference from a human, given that much of the process of its development would exactly mimic that of natural human (or other animal) development; thus these types of human made biological systems would exhibit almost all (all?) of the physical correlates of the soul, but would have be made by artifice.
To deny the possibility of strong AI requires some sort of transcendent soul that is not strongly coupled to material, with all of the baggage that that entails (e.g. how/why do souls interact with matter at all)?
I'm using soul to indicate whatever the non-material spiritual essence(s) arise in a dualistic theory. I'm not trying to indicate that it is necessarily a personal essence.