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There are two famous cases in classical mechanics that fail to be deterministic.

The first, and most famous, is Norton's Dome, which corresponds to a system with a force of the form

$$F = \sqrt{r} $$

There are more details on the Wikipedia article (it's usually described as the result of a reaction force from a surface with a certain shape), but the basic idea is that the derivative of the force fails to be defined at $r = 0$, since

$$(\sqrt{r})' = \frac{1}{2\sqrt{r}}$$

Due to this, there's no guarantee that the equation $\ddot{r} = \sqrt{r}$ has a unique solution (and indeed it doesn't), because it fails to be Lipschitz continuous.

There's plenty of informations about Norton's Dome, both here and on the internet, so here's the more interesting, if even more pathological, example, the Space Invader.

The space invader is a particle which is submitted to an unbounded acceleration in finite time, so that it reaches "infinity" after a while. The exact form of the force doesn't matter, but for instance you could pick

$$F = \tan(t)$$

In such cases, the particle will go off to infinity at $t = \pi/2$ and, after that time, cease to exist. As this system is time-symmetric, it is also possible to consider the case of a particle which originally does not exist and comes from infinity, or even doing both (restriction of the force to specific time intervals will do to insure those outcomes).

Another example of such a behaviour are the Painlevé non-collision singularities. The most famous example of which is a 5-body gravitational problem where one of the particle will also go to infinity in finite time, by simply borrowing energy from two 2-body systems. As for point-particles, the potential energy is unbounded from below (since it is $E \propto -1/r$), it is possible to have an infinite kinetic energy while maintaining conservation of energy, by having the 2-body systems in it collapse.

For a general treatment on the issue of determinism in classical physics, you can also check this article of Earman, for instance.