I know that there are problems that cannot be solved by any algorithm, such as the Halting problem. I also know that some processes cannot be even adequately approximated by any Turing Machine (equivalently, any digital computer), meaning that some property of the process cannot be simulated due to its intrinsic nature. An example of this would be Chaos, and its non-periodicity.
My question is: are there any other interesting processes in nature that really are outside the realm of what Turing Machines / digital computers can tackle? Any other interesting problems (outside from the well-known textbook ones) that Turing Machines cannot solve?
The reason why I ask for this is that I'm teaching to computer science students, and I want to make sure they understand that computers are not all-powerful machines, but there are problems and phenomena that lie fundamentally outside the realm of possibilities of any digital computer. What powerful examples could I use, other than Chaos and the Halting problem, to support my argument?