How can we know that the decimal expansion of an irrational number will never repeat?

If there appears a period in the decimal expansion of a number then You have after subtracting a rational term an expression of the form
$\sum _{n=k}^{\mathrm{\infty }}(d_1...d_r){10}^{-rn}$
where $d_j\in \{0,...,9\}$ are the digits in the period and r its length. Clearly this expression is rational if and only if $\sum _{n=0}^{\mathrm{\infty }}(d_1...d_r){10}^{-rn}$ is rational and
$\sum _{n=0}^{\mathrm{\infty }}(d_1...d_r){10}^{-rn}=\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{d_1...d_r}{{10}^r}\right)}^n=\frac{1}{1-\frac{d_1...d_r}{{10}^r}}$
is rational as a geometric series. Note $\frac{d_1...d_r}{{10}^r}<1$. Since there are (quite sophisticated) proofs that π is irrational (even transcendental, i.e. no root of any polynomial with rational coefficients,) one concludes that its decimal expansion cannot lead to any period.