Produce the most digits of an irrational number with only 64 KB of memory
Python - 10157831 digits

import sys

sys.stdout.write("0.")
a = 0
while True:
    a += 1
    sys.stdout.write(str(a))


Prints 0.1234567891011121314151617181920..., which is the Champernowne constant in base 10, until the program runs out of memory.

When the program's counter a reaches a specific number x, it has printed out at least x digits (as every number has at least one digit), and more on the order of x log x digits. For x = 1010000, the binary data required to represent a would be about 4.1 kilobytes, which is well below the allotted limit.

Interesting to note - any computer running this program would run out of total memory long before it ever printed off even 10100 digits (let alone 1010000), due to there only being about 1080 particles in the observable universe and only about 1021 bytes of memory available for storage on all of Earth's data computing devices.

Wolfram Alpha says that the total number of digits printed is actually about 4.09776347957... × 10157831, assuming we can get all the way to 2524288 - 1 without running into memory issues with the bignum storage.