1. **State the problem:** Evaluate the expression $\left(\log_3 27 + (5^2)^2\right) - \log_3 9$.\n\n2. **Recall logarithm rules and exponentiation:**\n- $\log_b a$ means the logarithm of $a$ with base $b$.\n- $\log_b (a^c) = c \log_b a$.\n- $a^{m^n} = a^{m \times n}$ if the exponent is nested.\n\n3. **Evaluate each part:**\n- $\log_3 27$: Since $27 = 3^3$, $\log_3 27 = 3$.\n- $(5^2)^2$: Calculate the exponent first: $5^2 = 25$, then $(25)^2 = 625$.\n- $\log_3 9$: Since $9 = 3^2$, $\log_3 9 = 2$.\n\n4. **Substitute values back into the expression:**\n$$\left(3 + 625\right) - 2$$\n\n5. **Simplify:**\n$$628 - 2 = 626$$\n\n**Final answer:** $626$