Consider $13$ marbles that are labeled with positive integers such that the product of all $13$ integers is $360$. Moor randomly picks up $5$ marbles and multiplies the integers on top of them together, obtaining a single number. What is the maximum number of different products that Moor can obtain?
1. **Factorization of 360**:
   We start by factorizing 360 into its prime factors:
   \[
   360 = 2^3 \cdot 3^2 \cdot 5
   \]
   This means that any integer factor of 360 can be expressed as \(2^a \cdot 3^b \cdot 5^c\) where \(0 \leq a \leq 3\), \(0 \leq b \leq 2\), and \(0 \leq c \leq 1\).

2. **Counting the Factors**:
   The total number of factors of 360 can be calculated by considering all possible combinations of \(a\), \(b\), and \(c\):
   \[
   (3+1)(2+1)(1+1) = 4 \cdot 3 \cdot 2 = 24
   \]
   Therefore, there are 24 different factors of 360.

3. **Labeling the Marbles**:
   To achieve the maximum number of different products, we need to label the marbles in such a way that we can form all 24 factors of 360 by picking any 5 out of the 13 marbles. One way to do this is to label five marbles with the factors \(2, 3, 3, 4, 5\) and the remaining eight marbles with 1. This ensures that any product formed by picking 5 marbles will be a factor of 360.

4. **Forming Different Products**:
   By choosing any 5 marbles out of the 13, we can form different products. Since the remaining 8 marbles are labeled with 1, they do not affect the product. The five marbles labeled with \(2, 3, 3, 4, 5\) can be combined in various ways to form all 24 factors of 360. For example:
   - Picking \(2, 3, 3, 4, 5\) gives \(2 \cdot 3 \cdot 3 \cdot 4 \cdot 5 = 360\)
   - Picking \(1, 1, 1, 1, 1\) gives \(1\)
   - Picking \(2, 3, 1, 1, 1\) gives \(2 \cdot 3 = 6\)
   - And so on.

5. **Conclusion**:
   Since we can form all 24 factors of 360 by picking any 5 marbles out of the 13, the maximum number of different products Moor can obtain is 24.

The final answer is \(\boxed{24}\)