1. **State the problem:** Evaluate the expression $16.66 - \left(\frac{1}{4} + 3.15\right)^3$. 2. **Identify the formula and rules:** We need to first simplify inside the parentheses, then cube the result, and finally subtract from 16.66. 3. **Calculate inside the parentheses:** $$\frac{1}{4} = 0.25$$ $$0.25 + 3.15 = 3.4$$ 4. **Cube the sum:** $$3.4^3 = 3.4 \times 3.4 \times 3.4 = 39.304$$ 5. **Subtract from 16.66:** $$16.66 - 39.304 = -22.644$$ 6. **Final answer:** $$\boxed{-22.644}$$