0.33333333... is also infinite. I'm pretty sure you wouldn't find my phone number in it though. Because pi is irrational it also has the property of being non repeating, but that's still not enough to contain whatever you are looking for. You could make an infinite, never repeating sequence without ever using the number 9 (by replacing all the 9s in the expansion of pi for example), in which case you wouldn't be able to find the year of birth of most monks.

It's actually unknown if Pi has the property of containing all possible strings, and while the currently known expansion does make it look so (pi looks normal), no matter how many digits are known, it won't ever be even a fraction of the infinite sequence.