Let's break down and solve the expression step by step:

The expression is:

$1 + \left[1 + \frac{1}{1 + \frac{1}{1 \cdot \left(1 + \frac{1}{2}\right)}}\right] + 1$

Step 1: Simplify the innermost expression $1 + \frac{1}{2}$:

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

Now the expression becomes:

$1 + \left[1 + \frac{1}{1 + \frac{1}{1 \cdot \frac{3}{2}}}\right] + 1$

Step 2: Simplify $1 \cdot \frac{3}{2}$:

$1 \cdot \frac{3}{2} = \frac{3}{2}$

Now we have:

$1 + \left[1 + \frac{1}{1 + \frac{1}{\frac{3}{2}}}\right] + 1$

Step 3: Simplify $\frac{1}{\frac{3}{2}}$:

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

Now the expression becomes:

$1 + \left[1 + \frac{1}{1 + \frac{2}{3}}\right] + 1$

Step 4: Simplify $1 + \frac{2}{3}$:

$1 + \frac{2}{3} = \frac{5}{3}$

Now the expression is:

$1 + \left[1 + \frac{1}{\frac{5}{3}}\right] + 1$

Step 5: Simplify $\frac{1}{\frac{5}{3}}$:

$\frac{1}{\frac{5}{3}} = \frac{3}{5}$

Now the expression becomes:

$1 + \left[1 + \frac{3}{5}\right] + 1$

Step 6: Simplify $1 + \frac{3}{5}$:

$1 + \frac{3}{5} = \frac{8}{5}$

Now the expression is:

$1 + \frac{8}{5} + 1$

Step 7: Add the constants:

$1 + 1 = 2$

So the expression becomes:

$2 + \frac{8}{5}$

Step 8: Convert to a common denominator:

$2 = \frac{10}{5}$

Now add:

$\frac{10}{5} + \frac{8}{5} = \frac{18}{5}$

So the final value of the expression is:

$\boxed{\frac{18}{5}} \quad \text{or} \quad 3.6$

Would you like more details or have any questions? Here are five related questions for you to consider:

1. How do we handle complex fractions within brackets?
2. How can the order of operations affect solving nested expressions?
3. What are other common ways to simplify fractions?
4. Why is it important to find a common denominator when adding fractions?
5. How would this process change with variables instead of numbers?

Tip: Always resolve parentheses and inner brackets first when dealing with nested operations.