Your question belongs to the context of *normal numbers*, which are real numbers whose digits and blocks of digits are uniformly distributed in the sense that each digits (or block of digits) occurs asymptotically with the correct "fair" frequency. The number you suggest is called "Champernowne's constant", it is known to be normal in base 10.

For a first introduction, see for example here: https://www.claymath.org/library/annual_report/ar2006/06report_normalnumbers.pdf

A number which is normal in all possible (integer) bases is called *absolutely normal*.

If the distribution of digits is not "fair", then the number is sometimes called "abnormal". See for example here: Martin, Greg. Absolutely abnormal numbers. Amer. Math. Monthly 108 (2001), no. 8, 746–754.

It is not known whether $\pi$, $e$, $\sqrt{2}$ etc. are normal in any base. However, it is conjectured that all algebraic irrationals are absolutely normal. See for example here for more information: Bailey, David H.; Crandall, Richard E.:
On the random character of fundamental constant expansions.
Experiment. Math. 10 (2001), no. 2, 175–190.