In one of Carl Sagan's Novels, the exact emerging unterminated value of pi is found to contain the plans for a time machine.

It seems to me that, since pi is irrational, it would not be exactly representable in any number system with any integer for a base.

It is true that a number might be a terminating sequence in one number base, and a repeating sequence in another. Consider the fraction 1/3 .
In base 10, this becomes 0.33333....
Yet in base 3 it would be 0.1

However 1/3 is by definition a rational number.

So a transcendental number like pi or e would not terminate in any integral number base.
If you used a number based on pi, I don't think you would be able to count very well.
MD