1. Model the Problem as a Graph:
Let $K_n$ be a complete graph representing the $n$ cities. In a complete graph, every pair of distinct vertices is connected by a unique edge. This means that every city is connected to every other city by a road.

2. Constraints on Misha's Travel:
Misha cannot visit a city more than once. Each time Misha travels from city $A$ to city $B$, the president destroys $k$ roads from city $B$, except the road Misha used to enter city $B$.

3. Determine the Maximum Number of Cities Misha Can Visit:
Assume Misha has visited $t$ cities. After visiting $t$ cities, Misha cannot continue if there are no more roads available to travel to a new city.

4. Calculate the Number of Destroyed Roads:
- Misha has traveled $t-1$ times (since she starts at the first city and travels $t-1$ times to visit $t$ cities).
- Each time Misha travels to a new city, the president destroys $k$ roads from that city.
- Therefore, the total number of destroyed roads is $(t-1) + k$.

5. Formulate the Inequality:
For Misha to be unable to continue her journey, the number of destroyed roads must be at least the total number of roads minus the roads she has used:
$$(t-1) + k \geq n-1$$
Simplifying this inequality:
$$t - 1 + k \geq n - 1$$
$$t + k - 1 \geq n - 1$$
$$t + k \geq n$$
$$t \geq n - k$$

6. Conclusion:
The maximum number of cities Misha can visit, regardless of how the president destroys the roads, is $n - k$.

The final answer is $\boxed{n - k}$.