My conclusion is that there is no difference between a number and an algorithm. Pi is not a "number" in a sense that integers and rationals are - it is not a finite string of symbols that represents a quantity for us. But it does represent some quantity, though in an indirect way - all irrational numbers can only be represented by an algorithm that approaches the infinitely-precise value.
So, integers and rationals only seem "like numbers" to us because we don't consider them "algorithms" - they seem intuitive to us. But looking from the perspective of Peano axioms, even integers are just algorithms for computing numbers - 3 is just s(s(s(1))), which is an algorithm that states that the successor function has to be applied 3 times to the number 1. So we can't really draw any kind of objective line between a number and an algorithm. Every number is an algorithm, it's just that some of them are trivial for us.
Same thing with circle - circle is just an algorithm for drawing a specific shape. All the properties of circles are just algorithms for approximating real-world properties of our drawings of circles.
So basically, pi (the algorithm) does in fact "contain" all sequences of digits when computed - but it does it in the same way that an infinite grassfield contains a grassleaf of any size - the size of the grassfield only makes it harder for us to find the grassleaf.
My conclusion is that there is no difference between a number and an algorithm. Pi is not a "number" in a sense that integers and rationals are - it is not a finite string of symbols that represents a quantity for us. But it does represent some quantity, though in an indirect way - all irrational numbers can only be represented by an algorithm that approaches the infinitely-precise value.
So, integers and rationals only seem "like numbers" to us because we don't consider them "algorithms" - they seem intuitive to us. But looking from the perspective of Peano axioms, even integers are just algorithms for computing numbers - 3 is just s(s(s(1))), which is an algorithm that states that the successor function has to be applied 3 times to the number 1. So we can't really draw any kind of objective line between a number and an algorithm. Every number is an algorithm, it's just that some of them are trivial for us.
Same thing with circle - circle is just an algorithm for drawing a specific shape. All the properties of circles are just algorithms for approximating real-world properties of our drawings of circles.
So basically, pi (the algorithm) does in fact "contain" all sequences of digits when computed - but it does it in the same way that an infinite grassfield contains a grassleaf of any size - the size of the grassfield only makes it harder for us to find the grassleaf.