A normal number [1] is pretty close to what you're describing. Although there are a lot of normal numbers, theoretically, they are very difficult to construct, and equally difficult to prove normality. Unfortunately, proving non-normality is also difficult, although the wikipedia article has some examples of base-10 normal numbers and a couple of examples of irrational non-normal numbers.
It's been 20 years since I majored in math, so maybe I did encounter these and just don't remember. Looks like there is a proof that almost all real numbers are normal, but where it comes to specific numbers pi, e, sqrt(2), they are thought to be normal but no proof.
A normal number [1] is pretty close to what you're describing. Although there are a lot of normal numbers, theoretically, they are very difficult to construct, and equally difficult to prove normality. Unfortunately, proving non-normality is also difficult, although the wikipedia article has some examples of base-10 normal numbers and a couple of examples of irrational non-normal numbers.
[1] https://en.wikipedia.org/wiki/Normal_number