If you want to calculate the volume of an $n$ dimensional sphere with radius $r$ you must include a power of $\pi^{n/2}$.

In general, we have $$V_n(r) = \dfrac{n\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}r^{n-1}$$

If we consider the ratio $R_n$ of the actual volume to the approximate volume we find that $$R_n = \frac{V_n(r)}{V_n(r)_{\text{approx}}} = \left(\dfrac{\pi}{\pi_{\text{approx}}}\right)^{n/2}$$

If we set $n = 2\cdot 40^{20}$ we find that $R_n \approx 1.10621$ rather than the expected value of $R_n = 1$.

This example is completely contrived but it demonstrates that the error will become apparent when large powers are involved. This is just a simple application of the use of significant digits.

There are many more examples using periodic functions with a period involving $\pi$. Consider finding

$$\tan(10^{100})$$

Essentially, you must find $10^{100}\! \pmod{\pi}$ which would require knowing $100$ digits of $\pi$ (to calculate the answer with some precision). You can find more information here. Again, this involves large powers requiring an excessive number of digits.

The easiest example of the need for an exact value most likely comes from numbers of an astronomical (or even extra-astronomical) scale. I don't know how convincing this argument would be to students and you can consider cross posting to MathEducators.

If you want an explanation for younger mathematicians you should explain how large powers of any number can an estimate unreliable and then present an applications of large powers of numbers or even just large powers of $\pi$. You can show how estimates like $\frac{1}{3} \approx .333$ are unreliable and then move up to more complicated examples.