You may have seen the claim that π contains every phone number, every birthday and the complete works of Shakespeare. It might be true. Here is what is actually known, and what is only very strongly suspected.
What is proved
π is irrational (Johann Lambert, 1761): its decimals never end and never settle into a repeating loop. It is also transcendental (Ferdinand von Lindemann, 1882): it is not the answer to any polynomial equation with whole-number coefficients, which is why squaring the circle with ruler and compass is impossible.
Neither fact says every number is in there. The number 0.101001000100001… never repeats either, and it will never contain a 2.
What is not proved
A number is called normal (in base 10) if every digit turns up a tenth of the time, every pair of digits a hundredth of the time, and so on for every length. A normal number contains every finite string of digits, infinitely often.
Nobody knows whether π is normal. Nobody has even proved that every digit appears infinitely often: as far as a proof goes, π might run out of 7s one day. Almost every number is normal (that much is proved), yet not one famous constant, not √2, not e, not π, has been shown to be.
What the digits say
Every test so far finds π's digits looking like the output of a fair ten-sided die. In the trillions of digits computed, each digit makes up a tenth of them to within the wobble you would expect from chance, and so do pairs, triples and longer runs.
If the digits do behave like dice, the arithmetic is simple. Each ten-digit window is one of 10 billion numbers, so a particular one is still missing after N digits with probability e^(−N/10¹⁰). After 100 billion digits that is e^(−10): 99.995% of all ten-digit numbers have turned up, and about 454,000 have not.
So is my number in there?
A typical ten-digit number first turns up about 7 billion digits in, and 99% of them have appeared by 46 billion. In the first 2 billion digits only 18% of them are there: about 8.2 billion of the 10 billion are still missing.
Longer numbers need far more digits: every extra digit makes the wait about ten times longer. An eleven-digit number typically needs ten times as many digits as a ten-digit one, and a sentence of Shakespeare is out of reach of every computer ever built. So when we can't find a long number, we show you its longest piece instead.
Try it: find your number, or see how the search works.