I have heard, but never done the math the verify, that with 50-ish digits of pi, one's error on a circle the size of the Universe would be smaller than a plank length.
Although this really should have been posted 4 days ago,
2pi x 3x10^8 x 40x10^10 / 1.6x10^-35
is just over 10^54 so that's about all the digits you need to memorize. Unless you're looking for a really strong password, reciting 100,000 digits is probably more than necessary:
http://blogs.scientificamerican.com/observations/how-much-pi...