1. **State the problem:** Simplify the expression $$5^{3 - \log_2 25} + 3^{2 - \log_3 3} - 4^{4 - \log_2 5}.$$\n\n2. **Recall logarithm and exponent rules:**\n- $a^{m-n} = \frac{a^m}{a^n}$\n- $\log_a a^k = k$\n- $\log_a b = c \implies a^c = b$\n\n3. **Rewrite each term using the exponent subtraction rule:**\n$$5^{3 - \log_2 25} = \frac{5^3}{5^{\log_2 25}}, \quad 3^{2 - \log_3 3} = \frac{3^2}{3^{\log_3 3}}, \quad 4^{4 - \log_2 5} = \frac{4^4}{4^{\log_2 5}}.$$\n\n4. **Calculate the simpler powers:**\n$$5^3 = 125, \quad 3^2 = 9, \quad 4^4 = 256.$$\n\n5. **Simplify terms with logarithms in the exponent:**\n- For $5^{\log_2 25}$, note $25 = 5^2$, so\n$$5^{\log_2 25} = 5^{\log_2 5^2} = 5^{2 \log_2 5}.$$\n- For $3^{\log_3 3}$, since $\log_3 3 = 1$,\n$$3^{\log_3 3} = 3^1 = 3.$$\n- For $4^{\log_2 5}$, keep as is for now.\n\n6. **Rewrite $5^{2 \log_2 5}$ using change of base:**\nSince $5^{2 \log_2 5} = (5^{\log_2 5})^2$, and $5^{\log_2 5} = 2^{\log_2 5 \cdot \log_5 5} = 2^{\log_2 5}$ (this is complex, better to use change of base):\n\nAlternatively, use the identity $a^{\log_b c} = c^{\log_b a}$:\n$$5^{\log_2 25} = 25^{\log_2 5}.$$\nSince $25 = 5^2$,\n$$25^{\log_2 5} = (5^2)^{\log_2 5} = 5^{2 \log_2 5}.$$\nSo this confirms the previous step.\n\n7. **Rewrite $4^{\log_2 5}$ similarly:**\nSince $4 = 2^2$,\n$$4^{\log_2 5} = (2^2)^{\log_2 5} = 2^{2 \log_2 5} = (2^{\log_2 5})^2 = 5^2 = 25.$$\n\n8. **Now rewrite the terms:**\n- $5^{\log_2 25} = 25^{\log_2 5}$ (from step 6)\n- $4^{\log_2 5} = 25$ (from step 7)\n\n9. **Substitute back into the expression:**\n$$\frac{125}{5^{\log_2 25}} + \frac{9}{3} - \frac{256}{25} = \frac{125}{25^{\log_2 5}} + 3 - \frac{256}{25}.$$\n\n10. **Evaluate $25^{\log_2 5}$ numerically:**\nSince $\log_2 5 \approx 2.3219$,\n$$25^{2.3219} = e^{2.3219 \ln 25}.$$\nCalculate $\ln 25 \approx 3.2189$, so\n$$e^{2.3219 \times 3.2189} = e^{7.478} \approx 1763.5.$$\n\n11. **Calculate the first term:**\n$$\frac{125}{1763.5} \approx 0.0709.$$\n\n12. **Calculate the last term:**\n$$\frac{256}{25} = 10.24.$$\n\n13. **Sum all terms:**\n$$0.0709 + 3 - 10.24 = 3.0709 - 10.24 = -7.1691.$$\n\n**Final answer:** $$\boxed{-7.1691}.$$