Unlikely, in its usual definitions, but I suppose some speculative theories do allow for it. I don't see how one could even recognize hypercomputation if it was staring them in the face - if we are Turing-emulable, we can't truly know if there are systems capable of operating with concrete infinities, at best we can theorize, but never know experimentally (we could also always be described by a computational, finite truncation of them at any given time, although never in full).
However, don't let that bother you, there are plenty of ways to achieve "unlimited" computational resources without having to violate the Church-Turing Thesis.
>>5
I really doubt consciousness and self-consciousness (different things) require hypercomputation. In some computationalist theory of mind (non-materialist ones; the materialist ones are forced to eliminate consciousness away as a delusion), consciousness is merely awareness that goes on at certain abstract structures contained within a lower-level computation, while self-awareness is a more specific structure/process of which one can be conscious of (and can be formalized mathematically).
However, if such a theory of mind predicts too many unusual, unstable experiences (which we don't have), we might have to assume some non-computationalist theories of mind which involve hypercomputation, however most of them seem utterly unplausible and evne deciding on a theory is difficult - the math everyone can agree on includes computational universality, but when you bring in concrete infinities, the theories start diverging greatly and there's no way for us to even know which one would be more likely, not to mention that there's zero evidence for any of them being required by any current theory of mind.
We are self-overcoming because we can change our beliefs, however that doesn't make us that rational, merely allows us the chance of being more correct the more we can update them and the better heuristics we choose to use.