A

Step 1: Understand the coefficient of restitution (e). The coefficient of restitution is defined as the ratio of relative velocities after and before an impact. When a ball bounces, its rebound height is reduced according to this coefficient.

Step 2: Establish the relationship between rebound height and coefficient of restitution. If a ball is dropped from a height $h$, the height after the first bounce ($h_1$) can be given by:
$$ h_1 = e^2 h $$
The height after the second bounce ($h_2$):
$$ h_2 = e^2 h_1 = e^2 (e^2 h) = e^4 h $$
The height after the third bounce ($h_3$):
$$ h_3 = e^2 h_2 = e^2 (e^4 h) = e^6 h $$
The height after the fourth bounce ($h_4$):
$$ h_4 = e^2 h_3 = e^2 (e^6 h) = e^8 h $$

Step 3: Therefore, the height of the ball after it rebounds for the fourth time is:
$$ h_4 = e^8 h $$

Conclusion: The correct answer for the height to which the ball goes after it rebounds for the fourth time is $e^8 h$. Hence, the height is indeed $h_3$ as specified in the question's relationship context.