The value of $$3\frac{5}{6} + \left[3\frac{2}{3} + \left\{\frac{15}{4}\left(5\frac{4}{5} \div 14\frac{1}{2}\right)\right\}\right]$$ is equal to:

First convert every mixed fraction in the expression to an improper fraction.

$$3\frac{5}{6} = \frac{3 \times 6 + 5}{6} = \frac{18 + 5}{6} = \frac{23}{6}$$

$$3\frac{2}{3} = \frac{3 \times 3 + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3}$$

$$5\frac{4}{5} = \frac{5 \times 5 + 4}{5} = \frac{25 + 4}{5} = \frac{29}{5}$$

$$14\frac{1}{2} = \frac{14 \times 2 + 1}{2} = \frac{28 + 1}{2} = \frac{29}{2}$$

Evaluate the innermost operation: $$5\frac{4}{5} \div 14\frac{1}{2}$$.

$$\frac{29}{5} \div \frac{29}{2} = \frac{29}{5} \times \frac{2}{29} = \frac{2}{5}$$

Next, multiply this result by $$\frac{15}{4}$$:

$$\frac{15}{4} \times \frac{2}{5} = \frac{30}{20} = \frac{3}{2}$$

Add this to $$3\frac{2}{3}$$:

$$\frac{11}{3} + \frac{3}{2}$$

Find a common denominator (6): $$\frac{11}{3} = \frac{22}{6}, \quad \frac{3}{2} = \frac{9}{6}$$

Sum: $$\frac{22}{6} + \frac{9}{6} = \frac{31}{6}$$

Finally add $$3\frac{5}{6} = \frac{23}{6}$$ to this result:

$$\frac{23}{6} + \frac{31}{6} = \frac{54}{6} = 9$$

Hence the value of the entire expression is 9.

Option C which is: 9