1. **State the problem:** Simplify the expression $$1 + \left\{ \frac{15}{4} - \left[ \frac{6}{8} \div \left( \frac{1}{2} + \frac{3}{4} \right) \right] \right\}$$. 2. **Recall the order of operations:** Parentheses, brackets, division and multiplication (from left to right), addition and subtraction (from left to right). 3. **Calculate inside the parentheses first:** $$\frac{1}{2} + \frac{3}{4} = \frac{2}{4} + \frac{3}{4} = \frac{5}{4}$$ 4. **Rewrite the division inside the brackets:** $$\frac{6}{8} \div \frac{5}{4} = \frac{6}{8} \times \frac{4}{5}$$ 5. **Multiply fractions:** $$\frac{6}{8} \times \frac{4}{5} = \frac{6 \times 4}{8 \times 5} = \frac{24}{40}$$ 6. **Simplify the fraction by canceling common factors:** $$\frac{24}{40} = \frac{\cancel{24}}{\cancel{40}} = \frac{6}{10} = \frac{3}{5}$$ 7. **Substitute back into the expression:** $$1 + \left\{ \frac{15}{4} - \frac{3}{5} \right\}$$ 8. **Find common denominator to subtract:** $$\frac{15}{4} - \frac{3}{5} = \frac{15 \times 5}{4 \times 5} - \frac{3 \times 4}{5 \times 4} = \frac{75}{20} - \frac{12}{20} = \frac{63}{20}$$ 9. **Add 1 to the result:** $$1 + \frac{63}{20} = \frac{20}{20} + \frac{63}{20} = \frac{83}{20}$$ 10. **Final answer:** $$\boxed{\frac{83}{20}}$$