As far as I know, current cosmological models predict an infinite universe. This leaves me wondering, given our current understand of cosmology:
Can a hypothetical cosmological-scale computer keep expanding it's memory and perform an unlimited amount of operations?
By expand I mean the device might send probes to nearby galaxies, and use resources from those galaxies to produce more probes to other galaxies. Also produced is some kind of memory device and a communication relay, allowing it to function as a "memory expansion".
Such a device would be a true physical realization of a Turing machine!*
Limits I can think of are the 2nd law of thermodynamics and of course the speed of light. But perhaps if this device expands faster than the decrease of the rate of computation due to
1) Increasing entropy
2) Increasing device radius (the communication cost per step should be proportional to intergalactic distance)
can it still perform arbitrarily many operations?
*Here's an explicit construction.
A Turing Machine consists of four elements: an infinite tape, a read/write head, a finite state register and a finite instruction table.
Each galaxy stores a constant $k$ amount of bits, thus galaxies function as a tape. The probe contains the state register and optionally also the instruction table and head. Thus we have a Turing Machine, provided the probe doesn't run out of energy.