What are real numbers called that have the following property: Given a non-terminating real number r and for any digit D, P(D is in the tail of r) > 0. I do not believe transcendental numbers have this properly but I could be wrong.
For example, one can easily prove interesting things about real numbers if one can show that any digit always exists.
For example if pi or e has these properties one can show that any subsequence of pi always exists in pi or e. (and hence a also as a sub-subsequence, etc)
The argument is quite simple and interesting. I think it demonstrates to the lay person how interesting numbers can be in that an infinite sequence can contain itself as a sub-sequence... it seems counter intuitive to the lay person but is quite natural for the type of numbers I'm describing.
for example, 0.01234567890123456790123456789....
has pi "embedded" in it as a subsequence and, in fact, as any sequence embedded in it. (pi, e, 2, or any other value)
Of course, the above value is somewhat contrived but it does demonstrate that we can, instead of thinking of "real numbers" and calculations on them, use sequences an and calculations on the sequences instead.
The property that you are describing sounds like a variation on the concept of "Normal Numbers". In a normal number every sequence of digits occurs with equal probability. A slightly broader concept is a disjunctive sequence which is a sequence that contains all possible subsqeuences (likewise a number who's decimal expansion is called a disjunctive number).
You might be able to say "The number is normal for strings of length n" or "The number is disjunctive for strings of length n".