Specifically, at the Planck scale, the smallest approximation to a circle would presumably involve 6 discrete, Planck-size "locations" or "space bits" ("metrobits"?) arranged around one central bit--thinking of them as blobs in a hexagonal close-packing arrangement. The diameter "center" to "center" would be 2. The circumference would be 6. Your "physical pi" would converge to an exact vallue of 3 at the ultimate limit.
Now I'm thinking - at one extreme "physical pi" approaches mathematical, ideal pi at the Hubble scale, while at the Planck scale it might well be exactly 3 if the above admittedly simplistic model is meaningful. So the difference (pi - 3, or .1416...) is potentially an interesting number. Like, could it relate to the fine structure constant somehow?
(Update) So after playing around with that last question I discovered the following tantalizing facts, using the fine structure approximation 1/137 (abbreviated FSA):
(pi - 3) / FSA = 19.3981...
Cube root of FSA = .193981...
Now... can that be a coincidence? OK. I know 1/137 isn't the real fine structure constant. But still...
Continuing the madness:
((pi - 3) / 100 FSA) [approximately =] cube root of FSA
FSA [approx =] fourth root of [(pi - 3) cubed / 1,000,000]
Hmmm.... fine structure constant just in terms of pi vs. 3... purely a geometric factor relating to the difference in actual Planck scale quantized path lengths vs. ideal continuous mathematical curves?