Given a certain number $\mathcal{N}$ with infinite digits, for instance $\pi$ or $\sqrt2$, it's possible to know if this number has a L-string of repeated numbers, but without actually computing the whole number $\mathcal{N}$?
To be concrete, take $\mathcal{N}=\sqrt2$. For instance I want to know if there is at least a sequence with $L=7$ times the number 3 (if yes, this means that $\sqrt2=1,41 \dots 3333333 \dots$), but without computing the infinite digits of $\sqrt2$.
Generally, no.
However, for a "naturally" occurring number, like those you show, everyone would be willing to bet yes, since it has been shown that almost all numbers are normal to every base.