$G$ is an undirected graph with $n$ vertices and 25 edges such that each vertex of $G$ has degree at least 3. Then the maximum possible value of $n$ is _________ .

According to the Handshaking Lemma, the sum of degrees of all vertices in an undirected graph equals twice the number of edges:

Given that the degree of every vertex is at least $3$, i.e., $\text{deg}(v) \ge 3$ for all $v \in V$:

Substituting the total sum of degrees:

For $n = 16$, we can have $14$ vertices of degree $3$ and $2$ vertices of degree $4$:

Since the number of odd-degree vertices is even ($14$), such a graph can exist.

Thus, the maximum possible value of $n$ is 16.