I don't see how it leads to a "god of the gaps" - can you clarify that?
A mathematical limit/approximation ... 3.14, 3.141, 3.1415, 3.14159 etc ... there is a limiting series of rational numbers, but you don't get Pi itself, that takes an infinite long decimal number. The difference between any rational number (as approximation) and its related algebraic (square root of two) or transcendental number (Pi) is always non-zero. Limits of course are also used in Calculus.
There is the argument, in atheism, of the "god of the gaps" ... that as naturalism is clarified and quantified, that we get closer and closer to the "truth". But in both cases, in a practical sense we can never arrive. Physics isn't limited to rational numbers. Not even limited to real numbers (see complex numbers). We accept on faith (if you are a lay person) that the final elimination of the "god of the gaps" is achievable after infinite effort. This happens all the time with hard math, even with real numbers (one dimensional). Many infinite series used in say classical physics, let alone quantum physics, don't converge nicely. Pi doesn't converge easily, but there are numerical formulas what are even worse.
Anyway, this really isn't about theology. In math, sometimes a real limit can be proven, but we don't have a means of calculating it (non-constructive proof), and sometimes we do have a means of calculating it (constructive proof). Pi falls into the later category. As a practical matter, having Pi to seven digits is sufficient for every engineering problem. Last month using computers, Pi is calculated to 9 trillion digits. For some nasty numbers (called normals) it is very hard to calculate even the first digit.
