A company has $n$ employees. It is known that each of the employees works at least one of the $7$ days of the week, with the exception of an employee who does not work any of the $7$ days. Furthermore, for any two of these $n$ employees, there are at least $3$ days of the week in which one of the two works that day and the other does not (it is not necessarily the same employee who works those days). Determine the highest possible value of $n$.
1. **Graph Representation and Vertex Coloring**:
   - Consider a graph where each vertex represents one of the \(2^7 = 128\) possible weekly schedules for an employee (each day can either be a workday or a non-workday).
   - Connect two vertices with an edge if their corresponding schedules differ by exactly one day. This means each vertex has a degree of 7.

2. **Vertex Coloring**:
   - Let \(v_1, v_2, \ldots, v_n\) be the vertices representing the weekly schedules of the \(n\) employees.
   - Paint each vertex \(v_i\) black.
   - Paint each of the 7 neighbors of each \(v_i\) white.

3. **Properties of the Coloring**:
   - No two black vertices are adjacent because if they were, their schedules would differ by exactly one day, contradicting the condition that any two employees' schedules differ by at least 3 days.
   - No vertex is painted white more than once. If a vertex \(v\) were painted white more than once, it would mean \(v\) is a neighbor of both \(v_i\) and \(v_j\). This implies \(v_i\) and \(v_j\) differ by at most 2 days, which contradicts the given condition.

4. **Counting Vertices**:
   - Each black vertex has 7 white neighbors, and there are \(n\) black vertices.
   - Thus, the total number of painted vertices is \(8n\) (since each black vertex and its 7 neighbors are distinct).
   - Since there are only 128 vertices in total, we have:
     \[
     8n \leq 128 \implies n \leq 16
     \]

5. **Constructing an Example with 16 Employees**:
   - We need to show that it is possible to have 16 employees with schedules satisfying the given conditions.
   - Consider the following 16 weekly schedules represented as 7-tuples of 0s and 1s:
     \[
     \begin{aligned}
     & (0,0,0,0,0,0,0), (1,1,1,0,0,0,0), (1,0,0,1,1,0,0), (1,0,0,0,0,1,1), \\
     & (0,1,0,1,0,0,1), (0,1,0,0,1,1,0), (0,0,1,1,0,1,0), (0,0,1,0,1,0,1), \\
     & (1,1,1,1,1,1,1), (0,0,0,1,1,1,1), (0,1,1,0,0,1,1), (0,1,1,1,1,0,0), \\
     & (1,0,1,0,1,1,0), (1,0,1,1,0,0,1), (1,1,0,0,1,0,1), (1,1,0,1,0,1,0)
     \end{aligned}
     \]
   - Verify that for any two of these schedules, there are at least 3 days where one employee works and the other does not. This can be checked by comparing each pair of schedules and ensuring they differ in at least 3 positions.

The final answer is \( \boxed{ n \leq 16 } \)