In a directed graph with $2013$ vertices, there is exactly one edge between any two vertices and for every vertex there exists an edge outwards this vertex. We know that whatever the arrangement of the edges, from every vertex we can reach $k$ vertices using at most two edges. Find the maximum value of $k$.
1. **Understanding the Problem:**
   We are given a directed graph (tournament) with 2013 vertices where there is exactly one directed edge between any two vertices. We need to find the maximum number \( k \) such that from any vertex, we can reach \( k \) other vertices using at most two edges.

2. **Initial Claim:**
   We claim that the maximum value of \( k \) is 2012. This means that from any vertex, we can reach all other 2012 vertices using at most two edges.

3. **Constructing Suitable Tournaments:**
   We start by constructing a suitable tournament on 3 vertices. Consider vertices \( a, b, \) and \( c \) with edges \( \overrightarrow{ab}, \overrightarrow{bc}, \overrightarrow{ca} \). In this tournament, from any vertex, we can reach the other two vertices using at most two edges.

4. **Generalizing the Construction:**
   If we have suitable tournaments \( T_a, T_b, \) and \( T_c \), we can construct a larger suitable tournament by combining these three tournaments and adding edges \( \overrightarrow{ab}, \overrightarrow{bc}, \overrightarrow{ca} \) for all \( a \in T_a, b \in T_b, c \in T_c \). This construction ensures that from any vertex in the combined tournament, we can reach all other vertices using at most two edges.

5. **Building Larger Tournaments:**
   Using the above method, we can build suitable tournaments with \( 3, 9, 15, 21, \ldots \) vertices. The general form of the number of vertices in these tournaments is \( 3 + 6m \).

6. **Verification for 2013 Vertices:**
   Since \( 2013 = 3 + 335 \cdot 6 \), we can construct a suitable tournament with 2013 vertices using the above method.

7. **Alternative Construction:**
   Another way to construct a suitable tournament is by adding two vertices \( x \) and \( y \) to an existing suitable tournament \( T \). We add the edge \( \overrightarrow{xy} \), all edges from \( T \), and edges \( \overrightarrow{tx}, \overrightarrow{yt} \) for all \( t \in T \). This method allows us to build suitable tournaments with \( 3, 5, 7, \ldots \) vertices, i.e., any odd number of vertices.

8. **Conclusion:**
   Since 2013 is an odd number, we can construct a suitable tournament with 2013 vertices using either method. Therefore, the maximum value of \( k \) is 2012.

The final answer is \( \boxed{ k = 2012 } \).