Eight football teams play matches against each other in such a way that no two teams meet twice and no three teams play all of the three possible matches. 
What is the largest possible number of matches?
1. **Understanding the Problem:**
   - We have 8 football teams.
   - Each team plays against every other team exactly once.
   - No three teams play all three possible matches among themselves.

2. **Graph Theory Representation:**
   - Represent each team as a vertex in a graph.
   - Each match between two teams is represented as an edge between two vertices.
   - The condition that no three teams play all three possible matches means that the graph should not contain any triangles (3-cycles).

3. **Applying Mantel's Theorem:**
   - Mantel's theorem states that the maximum number of edges in a triangle-free graph with \( n \) vertices is given by:
     \[
     \left\lfloor \frac{n^2}{4} \right\rfloor
     \]
   - For \( n = 8 \):
     \[
     \left\lfloor \frac{8^2}{4} \right\rfloor = \left\lfloor \frac{64}{4} \right\rfloor = \left\lfloor 16 \right\rfloor = 16
     \]

4. **Constructing the Graph:**
   - The only configuration that realizes the maximum of 16 matches is the complete bipartite graph \( K_{4,4} \).
   - In \( K_{4,4} \), the 8 vertices are divided into two groups of 4 vertices each.
   - Each vertex in one group is connected to every vertex in the other group, and there are no edges within the same group.

5. **Verification:**
   - In \( K_{4,4} \), there are \( 4 \times 4 = 16 \) edges.
   - Since it is a bipartite graph, it is naturally triangle-free, satisfying the condition that no three teams play all three possible matches.

Conclusion:
\[
\boxed{16}
\]