Find the maximum possible number of edges of a simple graph with $8$ vertices and without any quadrilateral. (a simple graph is an undirected graph that has no loops (edges connected at both ends to the same vertex) and no more than one edge between any two different vertices.)
1. **Understanding the problem**: We need to find the maximum number of edges in a simple graph with 8 vertices that does not contain any quadrilateral (4-cycle).

2. **Applying Istvan Reiman's theorem**: According to Istvan Reiman's theorem, the maximum number of edges \( E \) in a simple graph with \( n \) vertices that does not contain a quadrilateral is given by:
   \[
   E \leq \left\lfloor \frac{n}{4} (1 + \sqrt{4n - 3}) \right\rfloor
   \]
   For \( n = 8 \):
   \[
   E \leq \left\lfloor \frac{8}{4} (1 + \sqrt{4 \cdot 8 - 3}) \right\rfloor
   \]
   Simplifying inside the floor function:
   \[
   E \leq \left\lfloor 2 (1 + \sqrt{32 - 3}) \right\rfloor
   \]
   \[
   E \leq \left\lfloor 2 (1 + \sqrt{29}) \right\rfloor
   \]
   Since \(\sqrt{29} \approx 5.385\):
   \[
   E \leq \left\lfloor 2 (1 + 5.385) \right\rfloor
   \]
   \[
   E \leq \left\lfloor 2 \cdot 6.385 \right\rfloor
   \]
   \[
   E \leq \left\lfloor 12.77 \right\rfloor
   \]
   \[
   E \leq 12
   \]

3. **Checking realizability**: Although the upper bound is 12, we need to check if it is possible to construct a graph with 8 vertices and 12 edges without forming a quadrilateral. 

4. **Constructing a model**: A known construction that avoids quadrilaterals and has 11 edges is as follows:
   - Vertices: \(\{a, b, c, p, q, r, x, y\}\)
   - Edges: \(\{ab, bc, ca, pq, qr, rp, ap, xb, xq, yc, yr\}\)

   This construction can be verified to ensure no quadrilateral is formed:
   - The subgraph formed by \(\{a, b, c\}\) is a triangle.
   - The subgraph formed by \(\{p, q, r\}\) is a triangle.
   - The edges \(\{ap, xb, xq, yc, yr\}\) connect the two triangles without forming any 4-cycles.

5. **Conclusion**: Since we can construct a graph with 11 edges that meets the criteria and cannot construct one with 12 edges without forming a quadrilateral, the maximum number of edges is 11.

The final answer is \(\boxed{11}\).