Abat hits a ball; which has the greater acceleration, the bat or the ball?

The ball.

Explanation: Right before the impact, the bat will have a bigger acceleration than the ball (assuming that the ball is trowed softly), and after the impact, there will be some energy of the bat transferred to the ball. As the ball has a lot less mass than the bat, it is "easily" accelerated:

Remember that the second Newton's law says that:

F = m*a

So, after the impact, the ball has a greater acceleration than the bat (Also, if the bat had more acceleration than the ball, it will "push" the ball, but you can see that the ball "escapes" to the bat after the initial impact).