I've heard many times people debate the possibility of a real world computation that is impossible for a Turing machine, especially in the context of a human mind. Implying that the Church-Turing thesis is wrong and that a Turing machine does not accurately model any possible real world computation. To me, it seems that not only is the Turing machine not too restrictive, it's actually unrealistically strong. Namely, it has unbounded tape and time in which to perform its computation!
I suppose that the counterargument to this would be that unbounded is not the same as infinite, and in an idealized model of the physical universe we really have unbounded space and time in which to perform computations (neglecting minor details like the expected heat death of the universe and the possibility of finite space). But still, it seems as though we are basing our ideas of what is computable on an unreasonably strong model.
So, are there any attempts to capture the notion of computability and what we know to be computable in the physical world in a model that is finite in nature? Or is there some deep inherent reason why unboundness is a requirement of any such model?
Edit: To address the comments so far - I know that a Turing machine with bound tape is equivalent to a DFA, and therefore definitely does not capture in any reasonable way the notion of possible real world computations. I was rather referring to models that might not resemble Turing machines at all.