No, it does not. It's possible for some string to repeat an arbitrarily long but still finite number of times, but the digits of $\pi$ cannot perfectly repeat after a certain point. If this were the case, then $\pi$ would be a fraction, which we know it's not. It will probably have strings that repeat any arbitrarily high finite number of times somewhere in it, but it will never repeat exactly past some point.
Say I'm trying to figure out the decimal expansion of some fraction, like $\frac56$. Let's call the decimal expansion $0.b_1b_2b_3b_4....$ where the $b_n$ are digits. Multiplying by $10$, we find that $\frac{50}6=8\frac26=b_1.b_2b_3b_4...$. Matching integer parts we find that $b_1=8$. Matching fractional parts we find that $0.b_2b_3b_4...=\frac26$. Multiplying by $10$ again gives us $b_2.b_3b_4b_5...=\frac{20}6=3\frac26$, so by similar reasoning $b_2$=3 and $0.b_3b_4b_5...=\frac26$. But look! Our new set of decimals is equal to $\frac26$ just like our last one! And we just computed that the first digit of $\frac 26$ (here $b_2$) was $3$, meaning that this new digit has to be three as well. But we already know that "subtracting three and multiplying by 10" will give us another $3\frac26$, meaning that these threes must continue indefinitely. So we've figured out that
$\frac56=0.8333333 . . .$
This always has to happen. Because the numerators can only take on values from $0-5$, and if we ever get the same numerator twice we know that the expansion will continue exactly like it did the last time we got that numerator. It had to repeat. You can reverse this process as well, so that any repeating decimal can be converted back into its fractional form. So $\pi$ not being a fraction is exactly the same as saying that $\pi$ never has a recurring decimal sequence.