The maximum degree of any vertex in a simple graph with n vertices is n - 1.

Explanation:
A simple graph is a graph that does not contain any loops (edges that connect a vertex to itself) or multiple edges (more than one edge between the same pair of vertices). In other words, each edge in a simple graph connects two distinct vertices.

In a simple graph with n vertices, each vertex can be connected to at most n - 1 other vertices. This is because a vertex cannot be connected to itself, so it can have at most n - 1 neighboring vertices.

To understand why the maximum degree of any vertex is n - 1, let's consider the worst-case scenario. In this scenario, each vertex in the graph is connected to every other vertex, creating a complete graph.

Key Points:
- A complete graph is a graph in which every pair of distinct vertices is connected by an edge.
- In a complete graph with n vertices, each vertex is connected to every other vertex.
- Therefore, the degree of each vertex in a complete graph is n - 1.

Example:
Let's consider a simple graph with 5 vertices. The maximum degree of any vertex in this graph is 4 (n - 1).

(2)------(3)
| |
| |
(1)------(4)
|
(5)

In the above example, each vertex is connected to at most 4 other vertices. Therefore, the maximum degree of any vertex is 4.

Conclusion:
In a simple graph with n vertices, the maximum degree of any vertex is n - 1. This is because each vertex can be connected to at most n - 1 other vertices.