1. **State the problem:** Find the real root of the cube root of 559, i.e., find $\sqrt[3]{559}$ rounded to the nearest thousandth. 2. **Formula and explanation:** The cube root of a number $x$ is a number $y$ such that $y^3 = x$. 3. **Calculate the cube root:** Using a calculator, compute $\sqrt[3]{559}$. 4. **Intermediate step:** The cube root of 559 is approximately 8.241. 5. **Rounding:** Round 8.241 to the nearest thousandth, which remains 8.241. 6. **Final answer:** $$\boxed{8.241}$$