A spigot algorithm pumps out digits one at a time and does not reuse the digits after they are computed. In 1995 Stanley Rabinowitz and Stan Wagon produced a remarkable spigot algorithm for generating decimal digits of π (paper in pdf). Today the remarkable Akira Haraguchi finished reciting 100,000 digits of π from memory – a world record.
You can use the Pi-Search Page to find the location of the digits of π within π. For example, The string 3 occurs at position 9 counting from the first digit after the decimal point. It looks like the following plotted on a log chart.