Will an ideal bouncing ball ever stop bouncing? By an ideal ball I mean a bouncing point mass with the property that it loses half of the velocity (50%) after each bounce. Now according to my definition of ideal ball, it is impossible for the ball to be at absolute rest once dropped from some height. That is because absolute rest would imply that ball has lost 100% of its velocity after say $N$th (it turns out that this number is infinity) collision which contradicts the definition. But calculating time by summing the time series does tend to a finite value (say 10seconds). Then my question is whether the ball will be at absolute rest after say 15seconds? What am I missing here which is causing contradiction between the answer (10s) and the definition of ideal ball?
 A: This is a mathematical idealisation. You would be correct that halving the bouncing velocity (say, peak velocity) at every bounce will never result in zero velocity mathematically.
But this is not a working physical idealisation. In physics, an ideal bouncing ball would lose no peak speed (or kinetic energy or bouncing height or whichever parameter it is measured by) per bounce. Otherwise you will have to account for where this lost energy disappears to if you still want it to somewhat resemble reality within the laws of nature. And the very reason that it would disappear might be due to dampening forces which bring us out of the idealisation and thus would eventually result in zero speed. So this seems to be an unrealistic idealisation to make for the a physics scenario in the first place. Rather, a physicist would argue that under ideal circumstances, the ball will forever bounce to the same height without ever slowing down.
A: There are two ways to look at this problem.
First, lets look at the problem mathematically, The contradiction comes due to the measurement of time. So, how do you measure time in an isolated system ? Because as soon as you bring an observer(clock), you are not in an ideal world. There is no way to measure time except counting the number of times the ball bounces. In a way, in an universe, where this idealisation is real, you would not see the ball stop. But, also, you would realise that total time is finite because time itself would be non-linear and measured by number of times the ball bounces.
Now look at the physical picture. Using coefficient of restitution is a way to model inelastic collision which is a way to account for loss in kinetic energy of ball.
This itself is an non-ideal situation because law of conservation of energy needs to be true. So, you can't see this as an isolated system. The kinetic enegy must have gone somewhere( increse in thermal vibration in molecules of ball and surface). The model works well in classical mechanics when length, time and energy scales are much large compared to atoms where atomic description becomes important. So, invoking the model where ball itself is made of subatomic particles becomes important.
A: 
Will an ideal bouncing ball ever stop bouncing?

Denote $t_0$ is the time it takes the ball to go from the first bounce to the next. Because it loses half its velocity with each bounce, each bounce takes half the time than did the previous bounce. Thus the total time spent bouncing is finite, even with an infinite number of bounces:
$$t_{\text{total}} = \sum_{n=0}^\infty \frac{t_0}{2^n} = 2t_0$$
Your perfect ball bounces an infinite number of times in a finite amount of time.

What am I missing here which is causing contradiction between the answer (10s) and the definition of ideal ball?

There is no contradiction. First off, your idealization violates physics at the sub-Planck level. At some time before hitting the point in time at which bounces become shorter than the Planck length, whether the ball is still bouncing is unmeasurable. Idealizations typically violate physics at some level. The answer is wrong if the violations are severe. If they're negligible, an idealization can still provide a meaningful answer.
The meaningful answer to your question is that after ten seconds (the example in the question), the ball is not bouncing. At some point prior to ten seconds, the ball is observably bouncing. In between those points, who cares? Physicists don't care. Richard Feynman famously disdained philosphy. It's your idealization that results in an infinite number of bounces in a finite amount of time.
A: The height of the N-th jump is going down exponentially in N. Very quickly this height will be much smaller that proton radius. The contradiction is that you cannot think of your ball as a continuous elastic object at these lengthscales.
