Let g(i) be the minimal Goedel sequence (generated by a suitable decimal encoding scheme x) in bijective correspondence with the computable real number n(i), so that the sequence g(1), g(2),... is in ascending order by length of its member sequences.
Informally, g(i) is the minimal (finite) length "Program" sequence defining representing the recursive function, which defines the value of n(i) with i being its corresponding Goedel number.
We define the number Ux as the number with the decimal expansion 0.g(1)g(2)..... We have thereby constructed a number in the domain [0;1] which encodes the set of all computable real numbers with an associated total order.
The real number Ux is clearly definable, non-random, and most certainly transcendental. But is it computable?
Throwing the question out there for inspiration...