The expression given is:

[ 1 + \left[ \left( \frac{3}{2} \right)^{3} \right]^2 : \left{ \left[ \left( \frac{3}{2} \right)^2 \right]^1 : \left( -1 + \frac{1}{3} \right)^4 \right} ]

Let's simplify this step-by-step.

1. Simplify $\left( \frac{3}{2} \right)^3$:

   $\left( \frac{3}{2} \right)^3 = \frac{3^3}{2^3} = \frac{27}{8}$

   Then square it:

   $\left( \frac{27}{8} \right)^2 = \frac{729}{64}$

   So, $\left[ \left( \frac{3}{2} \right)^3 \right]^2 = \frac{729}{64}$.

2. Simplify $\left( \frac{3}{2} \right)^2$:

   $\left( \frac{3}{2} \right)^2 = \frac{9}{4}$

   Then raise it to the power of $1$ (which leaves it unchanged):

   $\left[ \left( \frac{3}{2} \right)^2 \right]^1 = \frac{9}{4}$

3. Simplify $-1 + \frac{1}{3}$:

   $-1 + \frac{1}{3} = -\frac{3}{3} + \frac{1}{3} = -\frac{2}{3}$

   Then raise it to the power of $4$:

   $\left( -\frac{2}{3} \right)^4 = \frac{16}{81}$

4. Evaluate the nested division inside the brackets:

   Now we have:

   $\frac{9}{4} : \frac{16}{81}$

   Dividing by $\frac{16}{81}$ is equivalent to multiplying by its reciprocal:

   $\frac{9}{4} \times \frac{81}{16} = \frac{729}{64}$

5. Final calculation:

   Now, substitute everything back:

   $1 + \frac{729}{64} : \frac{729}{64}$

   The division $\frac{729}{64} : \frac{729}{64}$ simplifies to $1$, so we get:

   $1 + 1 = 2$

Final Answer:

The expression evaluates to:

$\boxed{2}$

Would you like further clarification on any step?

Related Questions:

1. What is the general process for simplifying nested exponents?
2. How does the order of operations apply to nested expressions with division?
3. Can this method be applied to different fractions with higher powers?
4. What if we replace $\frac{3}{2}$ with another fraction? How would that affect the result?
5. How do negative numbers behave when raised to even powers?

Tip: When simplifying complex expressions, always handle exponents and brackets carefully to maintain the correct order of operations.