I suppose that you could store information by the position of a particle
in a box, and then if you could make the box twice as big, you could
store an extra bit of information (not a perfect example, due to the
momentum uncertainty, but maybe there are nondemolition measurement
systems which would work better).
In The Physics of Immortality, Frank Tipler argues that open universes
impose limits on calculation. Unfortunately his argument is not technical.
Here is what he says, on page 139 (chapter 3, section "Experimental Tests
of the Omega Point Theory"):
"As I discussed in Chapter III, Freeman Dyson pointed out that, although
the energy is available in open and flat universes, the information
processing must be carried out over larger and larger proper volumes.
This fact ultimately makes impossible any communication between opposite
sides of the 'living' region in a flat universe, because the redshift
implies that arbitrarily large amounts of energy must be used to signal,
and Dyson showed that only a finite amount of energy is availble.
On the other hand, open universes expand so fast in the far future that
it becomes impossible for structures to form of sufficiently larger and
larger size to store a diverging amount of information."
Most of Tipler's book is devoted to exploring how things would go in a
closed universe, and he doesn't go into much detail about exactly what the
problems are in an open universe. He doesn't refer to any technical
references to buttress the argument above, and it doesn't seem to be
mentioned in the "appendix for scientists".
I don't completely follow what Tipler's point is about redshifts.
I don't think he is saying that signalling involves a loss of energy,
since the photon used to signal can be recaptured, and although it will
be redshifted, the universe has cooled by the same factor so the photon's
energy upon recapture should be just as useful as the energy which was
used to create it.
He may be saying that at some point there won't be enough energy in
the whole universe to create a single photon of enough energy to signal
from one side of the computer to the other. Apparently there is only a
finite amount of energy in the (accessible?) universe. But I'm not sure
that there is a threshold energy which would be necessary to create the
signalling photon; perhaps photons can have arbitrarily small energies,
as long as they are hotter than the background.