To solve this problem, we need to determine how many distinct ways the child can build a wall with $n$ identical cubes. Each cube can be placed in such a way that it shares a face with the previous cube. This can be done either by placing the new cube on the ground or on top of the previously placed cube.

Let's analyze the process:

1. Understanding the Cube Placement: When starting with the first cube, there are no choices; it must be placed on the line.

2. Choices for Subsequent Cubes: For each subsequent cube, the child has two choices:
- Place it directly next to the previous cube on the same level (ground level).
- Place it on top of the previous cube.

3. Recursive Formulation:
- After placing the first cube, each additional cube requires a decision to be made independently of the previous decisions, except where cubes are supported.
- Hence, for each cube from the second to the $n$-th, there are 2 independent choices to be made.

4. Counting Distinct Structures:
- This leads to a binary decision problem for each cube position, starting from the second one.
- Therefore, there are $2^{n-1}$ distinct ways to arrange the $n$ cubes.

5. Conclusion: For $n$ cubes, the number of distinct walls is determined by the number of binary choices we make for the position of each cube starting from the second cube, which equates to $2^{n-1}$.

Thus, the number of distinct walls that can be constructed is:
$$\boxed{2^{n-1}}$$