Pi is far more than just the ratio of the diameter to the circumference, even as a constructivist: https://affinemess.quora.com/What-is-math-pi-math-and-while-...

It's been estimated that if the universe were a computer, it could have performed no more than 10^120 operations on 10^90 bits of data so far (based on the size, age, and total energy of the known universe). http://arxiv.org/abs/quant-ph/0110141 I think the number of physically relevant bits of pi would be represented in there somewhere. But there's a long road ahead. If the universe keeps "computing" forever, the precision of numbers involved could also keep growing.

Pi has other uses. The digits are real, but they just don't matter in practical engineering.

But lets assume that there really are 10 dimensions - in that case a volume of a 10 dimensional sphere will require (pi^5)*r^10.

If you want to measure the volume to 1 plank 10 dimensional cube, you will need more digits.

Pi appears in many more places than in ratio of circumferences to their diameter. For example, if you flip a coin n times, the probability of getting exactly n heads and n tails asymptotically tends towards 1 / sqrt(π*n). The probability that two randomly chosen integers are coprime is 6/π^2. Etc.

There is a list of formulae where π appears (some related to circles, some not) here: https://en.wikipedia.org/wiki/List_of_formulae_involving_π

But pi is not an empirical constant. It is a mathematical constant, with well-defined formulas such as pi/4 = 1-1/3+1/5-1/7+...