An unphysically large finite computer is one that's vastly larger than anything that could possibly fit into our universe, if the character of physical law is anything remotely like it seems to be.
We might be able to get a googol () computations out of this universe by being clever, but to get computations would require outrunning proton decay and the second law of thermodynamics, and operations () would require amounts of computing substrate in contiguous internal communication that wouldn't fit inside a single Hubble Volume. Even tricks that permit the creation of new universes and encoding computations into them probably wouldn't allow a single computation of size to return an answer, if the character of physical law is anything like what it appears to be.
Thus, in a practical sense, computations that would require sufficiently large finite amounts of computation are pragmatically equivalent to computations that require hypercomputers, and serve a similar purpose in unbounded analysis - they let us talk about interesting things and crisply encode relations that might take a lot of unnecessary overhead to describe using small finite computers. Nonetheless, since there are some mathematical pitfalls of considering infinite cases, reducing a problem to one guaranteed to only require a vast finite computer can sometimes be an improvement or yield new insights - especially when dealing with interesting recursions.
An example of an interesting computation requiring a vast finite computer is AIXI-tl, or Andrew Critch's parametric bounded analogue of Lob's Theorem.