Since π is irrational, the number of digits is infinite and we will calculate even more digits in the future. Beyond a few tens of digits or so, the exercise is done mostly to improve our computational techniques and to test hardware reliablity (and of course, some bragging rights.) Engineering calculations only need at most 16 digits to cover all conceivable projects on earth. All intergalactic calculations in the future should not require more than 64 digits or so.
Wikipedia has a chronology of the π computation and I've plotted the number of digits vs. year below. Note the log scale of the vertical axis.
There seem to be three phases of techniques/technologies involved in the computations of π throughout the history. The first part is from a few thousands years ago until 1400 A.D. or so where calculations were done mostly with geometric figures. The second part is from 1400 to 1900 A.D. where calculations were done manually using infinite series. The third part is after 1946 where electronic computers and fast-converging series were used. The three parts correspond to the three different slopes of the plot above.
After 1946, with electronic computers, the number of digits seems to grow approximately exponentially. This might reflect the exponential advance in computer speed since a small number of infinite series families were used in the computations. Here's the plot after 1946:
For more interesting info, check out Wikipedia's page on the history of π calculations.