A simple irrational number with the same first 11,667,755 digits as $\frac{2}{3}$
Solution 1:
The reason for the near-periodic behavior of the choice $x = 10^5 + \frac{9}{250}$ has to do with the fact that $$\frac{x}{\pi} \approx 31831.00007753496977,$$ which is almost an integer with error $\epsilon \approx 0.000077 < 10^{-4}$. Moreover, $$\frac{1}{\epsilon} \approx 12897.4062021729,$$ and now you can see why this many terms are needed.
The above also suggests that if you can find some choice of $x$ such that $$\frac{x}{\pi} - \left\lfloor \frac{x}{\pi} \right\rfloor$$ is extremely tiny, you can make this phenomenon extend to as large of a value of $k$ as you please. It just so happens that the particular choice $10^5 + \frac{9}{250}$ is also close to a round number in base 10.