To find the greatest positive integer $k$ that satisfies the partition property, we must ensure that the positive integers can be divided into $k$ subsets $A_1, A_2, \ldots, A_k$ such that for all integers $n \geq 15$ and for each $i \in \{1, 2, \ldots, k\}$, there are two distinct elements in $A_i$ whose sum is $n$.

Let's analyze the problem:

1. Understanding the Partition Requirement:
- Each subset $A_i$ should contain two distinct elements whose sum equals $n$ for every $n \geq 15$.
- This requires diversity in each subset so that various sums $n$ can be obtained by choosing two elements from any subset.

2. **Finding Constraints on $k$**:
- If $k$ is too large, it might not be possible to achieve the necessary sums with the limited numbers available in smaller subsets.
- If the number of subsets $k$ is small enough, each subset can incorporate a sufficient range of numbers to meet the summing requirement.

3. **Demonstrating a Working Value of $k$**:
- For $k = 3$, consider three subsets:
$$A_1 = \{ 1, 4, 7, 10, \ldots \} = \{ 1 + 3t \mid t \in \mathbb{Z}^+ \},$$
$$A_2 = \{ 2, 5, 8, 11, \ldots \} = \{ 2 + 3t \mid t \in \mathbb{Z}^+ \},$$
$$A_3 = \{ 3, 6, 9, 12, \ldots \} = \{ 3 + 3t \mid t \in \mathbb{Z}^+ \}.$$
- These sets distribute the positive integers cyclically into three groups based on their remainder modulo 3.
- For any integer $n \geq 15$, it can be verified that there exist two numbers in each subset whose sum equals $n$. For instance:
- Choose distinct integers $a = 3m + r$ and $b = 3n + r$ with $r = 1, 2, 3$ for subsets $A_1, A_2,$ and $A_3$, respectively.

4. **Proving $k > 3$ Does Not Work**:
- Suppose $k = 4$. Then we would need to find a regular way to partition the integers into four subsets while maintaining the sum property for each subset.
- However, constructing such a distribution generally fails for larger $k$ because the need to utilize higher integers to achieve every possible sum $n \geq 15$ becomes impractical.

Therefore, the largest value of $k$ that permits the construction of such a partition is:

$$\boxed{3}$$
This solution satisfies the conditions of the problem, ensuring that every required sum can be found by adding two distinct elements from each subset of the partition.