For the moment, I think we can limit ourselves to the number of atoms in the solar system. One rough estimate is that there are 10e29 stars in the universe. If the atoms were divided up approximately evenly between these star's systems, then there'd be 10e82/10e29=10e53. So we have one IPv6 address for each cluster of 10e15 atoms.
Except! I've heard it estimated that about half the matter in the solar system is in the sun, and we don't want to use up the sun to build computers, because we need it to power the computers. So, 10e14 atoms per IPv6 left to work with.
So the question before the audience is:can you build a device that implements an IPv6 stack and a minimal radio transmitter that allows it to communicate with other, similar devices, using only 10e14 atoms? If so, or if it can be done in less, then we may have a problem*. Otherwise, I think we should be fine for now.
(To give you a rough estimate of what you're working with:10e14 atoms of silicon would mass about 46 nanograms.)
Submit your solutions to firstname.lastname@example.org
* Although the problem may not be manifest until we convert the *entire* Earth, core and all, into these devices, along with all the other planets, and colonize the Oort cloud, and do the same there.