Are there any examples of non-computable real numbers?

I haven't thought this through, but it seems to me that if you let $BB$ be the Busy Beaver function, then $$\sum_{i=1}^\infty 2^{-BB(i)}=2^{-1}+2^{-6}+2^{-21}+2^{-107}+\ ... \ \approx 0.515625476837158203125000000000006$$ should be a noncomputable real number, since if you were able to compute it with sufficient precision you would be able to solve the halting problem.

Chaitin's constant is an example (actually a family of examples) of a non-computable number. It represents the probability that a randomly-generated program (in a certain model) will halt.

It can be calculated approximately, but there is (provably) no algorithm for calculating it with arbitrary precision.

Any language can be turned into a number, by setting the $i^{th}$ decimal to 1 if the $i^{th}$ word is in the language, and to 0 otherwise. So we can build for instance the number $H$, which describes the halting problem and is therefore uncomputable.