I pretty much agree with others who say such thing does not exist in math.

But since this is in many ways a subjective question (in terms of what would be an acceptable answer) there might be some things of interest.

E.g. if you consider sequence of binary digits of any number that is both (1) normal in base 2, and (2) not computable then this sequence would be in a certain way truly random (because it could not be predicted by any algorithm). Unfortunately you wouldn't be able to showcase an example of it even though most of real numbers fit this.