Successive
digits of pi
do form a predictable
sequence (otherwise we wouldn't be able to calculate them), but it is not a
stable one in the sense of eventually becoming repetitive.
The reasons are simple enough: pi is irrational, as proved here, whereas all decimals which display infinitely repeating patterns are rational. In fact, given a repeating decimal, any high school student can recognise it as representing the sum of a geometric progression and, using the formulas given here, calculate exactly which rational number it corresponds to. Therefore we know that the decimal expansion of pi never reaches a point where some string of digits is repeated forever.