A similar question is 'Is pi a
Normal number ?'
Here is an
interesting quote:
"Determining if numbers are normal is an unresolved problem. It is not even known if fundamental mathematical constants such as pi (Wagon 1985, Bailey and Crandall 2003), the natural logarithm of 2 ln2 (Bailey and Crandall 2003), Apéry's constant zeta(3) (Bailey and Crandall 2003), Pythagoras's constant sqrt(2) (Bailey and Crandall 2003), and e are normal, although the first 30 million digits of pi are very uniformly distributed (Bailey 1988)."
Though one could imagine a non-normal number that does have every possible series of digits within it. For example, concatenating the digits of every number one after the other, and doing it twice if it contains only 1s, is a non-normal number containing every possible series of digits.
However, I would expect every 'non-constructed, transcendental' normal number to also contain all subsequences in it.