It seems to me that most real numbers can't be calculated by any finite set of instructions, even if we make use of non-computable functions.
First, assume that we have an oracle that will provide the digits of any specified non-computable number to arbitrary precision. So, for example, if we state a particular instance of Chaitin's constant, then our oracle can gives us this value to arbitrary precision. This would allow us to perform operations on non-computable numbers. And by extension, we could even specify non-computable operations on non-computable numbers, so long as the result is also some real number, which our oracle can then provide to arbitrary precision.
Any such specification will consist of a finite set of instructions, some of which could include non-computable functions, the outputs of which will be provided by our oracle.
Note that the set of all such instructions is countable, since any language in which we express the instructions will necessarily be finite. It follows that even if we allow for the use and resolution of non-computable functions, we can only specify a countable subset of the reals using any particular language.
It follows that not only are most of the real numbers non-computable, they're incapable of articulation using a finite language.
Note that this is distinct from defining a particular real number. Instead, I am pointing out that even if we make use of non-computable functions, it does not seem possible to construct a one-to-one map from (A) the set of statements that define a computation to (B) the set of real numbers.
As a result, it does not appear to be possible to calculate the set of real numbers by making use of a finite set of non-computable functions.
In short, most real numbers can't even be calculated using a finite set of non-computable functions.
Is there a known set of functions that are capable of generating all real numbers?