Since computer programs are countable but real numbers are not, it follows that there must exist real numbers that cannot be calculated by any program. Unfortunately, by definition, there isn't an easy way of describing any of them!
In fact, there are many tasks (not just calculating real numbers) that computers cannot perform. The most well-known is the halting problem, the busy beaver problem is less famous but just as fascinating.
["The Turing Omnibus", A.K. Dewdeney].