1. **State the problem:** We need to find which of the given numbers has the greatest value among $4.\overline{2}$, $\sqrt{24}$, $\pi + 1$, and $\frac{9}{2}$.\n\n2. **Convert each value to a decimal approximation:**\n- $4.\overline{2}$ means $4.2222\ldots$ (a repeating decimal).\n- $\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$. Since $\sqrt{6} \approx 2.449$, then $\sqrt{24} \approx 2 \times 2.449 = 4.898$.\n- $\pi + 1 \approx 3.1416 + 1 = 4.1416$.\n- $\frac{9}{2} = 4.5$.\n\n3. **Compare the decimal values:**\n- $4.\overline{2} \approx 4.2222$\n- $\sqrt{24} \approx 4.898$\n- $\pi + 1 \approx 4.1416$\n- $\frac{9}{2} = 4.5$\n\n4. **Determine the greatest value:**\n$4.898 > 4.5 > 4.2222 > 4.1416$.\n\n**Final answer:** $\sqrt{24}$ has the greatest value.