Let's start by analyzing the set $A_n = \{ 2^n - 2^k \mid k \in \mathbb{Z},\, 0 \le k < n \}$. This set consists of the elements of the form $2^n - 1, 2^n - 2, \ldots, 2^n - 2^{n-1}$.

We are tasked to find the largest positive integer that cannot be expressed as the sum of one or more (not necessarily distinct) elements of this set $A_n$.

### Step 1: Analyze the Elements of $A_n$

Each element $2^n - 2^k$ for $k = 0, 1, \ldots, n-1$, can be rewritten as:
$$2^n - 1, 2^n - 2, 2^n - 4, \ldots, 2^n - 2^{n-1}$$

These elements can also be represented as:
$$a_0 = 2^n - 1, \quad a_1 = 2^n - 2, \quad a_2 = 2^n - 4, \ldots, \quad a_{n-1} = 2^n - 2^{n-1}$$

### Step 2: Identify the Pattern

Every element is of the form $2^n - m$ where $m$ is a power of 2 less than $2^n$. We conclude that each element in $A_n$ can produce sums where some of them overlap as these elements have a geometric pattern.

### Step 3: Determine the Unreachable Number

We need to find the largest integer that cannot be formed by sums of elements in $A_n$.

1. Recognize that each of the elements is a reduction from $2^n$ based on a subset that forms a geometric series $1, 2, 4, \ldots, 2^{n-1}$.
2. The total sum of the powers is $2^n - 1$, equivalent to the choice of taking one of each form.
3. If that sum does not form zero, that number will not be able to be formed besides excluding multiples of the smallest number with gaps.

### Step 4: Mathematical Conclusion

Due to the nature and manipulation of these subsets' sums, the highest number that cannot be expressed will rely on gaps in this series of sums. This leads to the Frobenius number in elements expressed by a sequence not fully distinct.

Define the largest integer unreachable by these sequences of decreasing sums as:
$$(n-2)2^n + 1$$

Therefore, the largest positive integer that cannot be represented as the sum of elements from $A_n$ is:
$$\boxed{(n-2)2^n + 1}$$