The problem is asking for the length of segment $AB$ based on the given conditions:

1. $D$ is the midpoint of segment $AC$.
2. $C$ is the midpoint of segment $DB$.
3. $AC = 3 \, \text{cm}$.

Step-by-Step Solution:

1. Interpretation of Midpoints:

   • Since $D$ is the midpoint of $AC$, it means: $AD = DC = \frac{AC}{2} = \frac{3}{2} = 1.5 \, \text{cm}.$
   • Now, $C$ is the midpoint of $DB$. So, we need to calculate $AB$.

2. Breaking $AB$ into segments:

   • $AB$ can be written as: $AB = AD + DB.$
   • From the first condition, we know that $AD = 1.5 \, \text{cm}$.

3. Relation between $DB$ and $DC$:

   • Since $C$ is the midpoint of $DB$, it follows that: $DC = CB.$
   • We already know that $DC = 1.5 \, \text{cm}$, which means: $DB = DC + CB = 1.5 + 1.5 = 3 \, \text{cm}.$

4. Final Calculation of $AB$: $AB = AD + DB = 1.5 + 3 = 4.5 \, \text{cm}.$

Conclusion:

The length of segment $AB$ is $\boxed{4.5} \, \text{cm}$.

Would you like further clarification on any step? Here are some related questions to explore:

1. How do you find the length of a segment when given midpoints?
2. What are the properties of midpoints in geometry?
3. How would the problem change if $AC$ had a different length?
4. Can you generalize this solution for any length of $AC$?
5. How do you apply the midpoint theorem in different geometric contexts?

Tip: Always remember that the midpoint divides a line segment into two equal parts. This concept is fundamental when solving geometry problems involving midpoints.