Given a regular polygon with $p$ sides, where $p$ is a prime number. After rotating this polygon about its center by an integer number of degrees it coincides with itself. What is the maximal possible number for $p$?
1. **Understanding the problem**: We need to find the largest prime number \( p \) such that a regular polygon with \( p \) sides coincides with itself after rotating by an integer number of degrees.

2. **Rotation symmetry of a regular polygon**: A regular polygon with \( p \) sides will coincide with itself after a rotation by \( \frac{360^\circ}{p} \) degrees. This is because the polygon has rotational symmetry of order \( p \).

3. **Condition for rotation**: For the polygon to coincide with itself after rotating by an integer number of degrees, \( \frac{360^\circ}{p} \) must be an integer. This implies that \( p \) must be a divisor of 360.

4. **Prime factorization of 360**: The prime factorization of 360 is:
   \[
   360 = 2^3 \cdot 3^2 \cdot 5
   \]
   The prime factors of 360 are 2, 3, and 5.

5. **Finding the largest prime divisor**: Among the prime factors of 360, the largest prime number is 5.

6. **Conclusion**: Therefore, the maximal possible value of \( p \) is 5.

\(\blacksquare\)

The final answer is \( \boxed{5} \)