The purpose is not to simply to get as many digits of pi as possibe for any calculation purpose. It's also intended to discover if pi is truly irrational, i.e. is not the ratio of any two integers. Also, I believe to prove pi is also a transcendental number, from Wkipedia:
"In mathematics, a transcendental number is a real or complex number that is not algebraic – that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best-known transcendental numbers are and e"
I think "known" in that statement is truly "assumed" at least case of pi and the purpose of calculating it to more and more digits is to discover if the assumption is correct: i.e. proof.
Performing such poofs for pi on paper is really slow in comparison to the reported experiment.
One use of an irrational number today is that any block of digits or sequence generated by exression (e.g. digitsm 3rd, 9th, 81st...) within it and of suitable length makes a useful cryptographic key since no other block of the same length contains the same digits in the same order. It's alredy been posted that a block of a few digits repeats three times, that's not sufficient to make it irrational and ony requires one extra digit to be a pattern that doen't repeat. So you can't just test if chunks of pi as keys will de-crypt data because your likelihood of guessing the key takes up to the same time as the reported experiment for any sequence and infinitely longer for a sequence of digit positions created by expression. As understand it.
Take a simple example, you want an 8 digit cryptographic key. Generate the sha256 checksum of two word you can remember (already the sha256 numbers will probably never be guessed). First sum the digits in the first sha256 and use it as a "base". Now take the square of each (or evey other) digit in the other sha256 as a "position" and use the digit of pi at at the base plus that position in pi 8 times to get your final key. Using pi to a few hundred digits you will get a key that no-one could hope to guess in any useful time other than by pure luck. But if pi is not irrational, multiple guesses would return the key by pure luck.