To find the number of unordered triples $(X, Y, Z)$ that satisfy $X \cup Y \cup Z = A$ where $A = \{1, 2, \ldots, n\}$, we need to consider the following steps:

1. **Counting the ways to assign each element to subsets $X$, $Y$, and $Z$:**
Each element in $A$ can be in any of the subsets $X$, $Y$, or $Z$, or in any combination of these subsets. There are $2^3 = 8$ possible ways to assign each element to the subsets $X$, $Y$, and $Z$ (since each element can either be in or out of each subset).

2. Excluding the empty assignment:
Out of these 8 possibilities, one possibility is that the element is not in any of the subsets $X$, $Y$, or $Z$. This is not allowed since $X \cup Y \cup Z = A$ must cover all elements of $A$. Therefore, we have $8 - 1 = 7$ valid ways to assign each element to the subsets.

3. Calculating the total number of assignments:
Since each of the $n$ elements in $A$ can be assigned to the subsets in 7 different ways, the total number of ways to assign all elements is $7^n$.

4. Considering unordered triples:
The problem asks for unordered triples $(X, Y, Z)$. However, since the assignment of elements to subsets is inherently ordered, we do not need to further adjust for ordering.

Thus, the total number of unordered triples $(X, Y, Z)$ that satisfy $X \cup Y \cup Z = A$ is $7^n$.

The final answer is $\boxed{7^n}$