Do numbers exist for which we can't write down a formula or approximation?
For example, although Pi is irrational there are numerous ways to approximate it's value, such as continued fractions, series representations, or rational numbers such as 22/7. Does such a number exist where we can't do this?
The example I was considering in my head is an (irrational?) number less than 1 where each digit uniformly distributed between 0 and 9. That is, the first decimal digit is U(0,9) the second is U(0,9), etc. In the mean over numerous trials I would expect this to coverge to .444444....