From what I remember back in school we were taught that “inexact numbers” were those that had repeating or otherwise infinite sequences. The name referred to the representation of the number (in base 10) rather than the number itself.
There is nothing “inexact” about pi. The inexactness is in any attempts to render pi in decimal form. Thus, you would not say that pi is inexact, but that decimal approximations to pi are inexact.
Not sure what you mean by “repeating otherwise infinite sequences.” A number like 1/3 can be expressed as an infinite but repeating decimal 0.33333… The repeat cycle may be long, like 0.123456712345671234567… but still, there is a pattern that repeats itself over and over, without variation, ad infinitam. Such numbers are called “rational” numbers, they can be expressed as a quotient of two integers.
A number like SQRT(2) is an infinite decimal that never repeats itself. Such numbers are called “irrational” and pi is such an one.
Pi has the additional property that it cannot be expressed as the solution of a (finite) polynomial equation. It is thus called “transcendental.” (SQRT(2) is a solution of the equation x^2 = 2 so is not transcendental.)