Examples of events that are unpredictable as a matter of principle?
Yes, Heisenberg's uncertainty principle in quantum mechanics says as the position of a particle becomes more constrained, it's momentum becomes more uncertain, and vice-versa. This is not simply the limits of our own observation, but a real property of all wave-like systems.
More formally stated, σx σp >= h / 4π. σx is the standard deviation (uncertainty) about position, σp for momentum, and h is the Planck constant.
So if you went back in time and ran the same quantum system over again, you'd get a different result.
Radioactive decay is another quantum system which, at the level of single atoms, it is impossible to know when it will decay no matter how long you've been observing the atom. An atom's half life is a statistical probability for a large group of atoms, but a single atom's decay cannot be predicted. If you watched an atom until it decayed, rolled time back, and watched it again, you could not predict when it would decay.
This resulted in the famous Schrödinger's cat thought experiment. Make a box with a vial of poison. There's a trigger which is waiting for the decay of a single atom. When that happens, the vial is broken and the cat is killed. Without opening the box, is the cat alive or dead? Schrödinger intended the experiment to point out the apparent absurdity of the Copenhagen interpretation of the uncertainty principle, but it's since become a succinct explanation of quantum superposition.
This is all according to most mainstream interpretations of quantum mechanics, the most mainstream being the Copenhagen interpretation. The Many-worlds interpretation is also popular, and it also says if you run time back and forward you'd get a different result (a different "world").
But others, like De Broglie–Bohm theory, try to reconcile experimental observation with a deterministic quantum world.
There is no consensus among quantum mechanics experts as to which interpretation is correct. Sean Carroll called the results of the poll "The Most Embarrassing Graph in Modern Physics". So while I wrote the answer with certainty... we're really not sure.
It's also important to point out that this applies at the level of individual particles. At macroscopic scales it all averages out, so unless you set up a specific experiment or apparatus to measure individual particles, like Schrödinger's cat, you'd see the same macroscopic result.
I am going to disagree with the answers posted so far. I believe quantum mechanics is in principle entirely deterministic: given the wave function of the universe at any moment, its time evolution is completely determined. The problem with experiments is that we can never know the exact wave function (right down to phase) of the universe, or even of a single electron. That is what makes the outcome of experiments unpredictable. In principle, if we knew ALL the relevant information (the exact wave function at any given moment) we could predict the outcome of any measurement.
DISCLAIMER: I am a recognized crackpot in this forum whose answers are routinely and massively downvoted by people who know much more than me.
I would argue as a matter of principle that all of reality is essentially unpredictable. That is, if you went back in time and witnessed the same universe go forward again, the outcome would start to differ, first minimally, and then increasingly so. The reason is that quantum processes are inherently unpredictable and some systems are non-linear to a degree that the small random events have large-scale repercussions. The proverbial Butterfly Effect is an example. Or just a sequence of very classical billiard ball collisions. After a finite number of collisions the quantum state of a single atom in the first ball is sufficient to significantly alter the last ball's direction. There is nothing one can do about it.
So what about astronomy? Movements of celestial bodies are entirely predictable, aren't they? Even Stone-age cultures could predict eclipses, and we can predict the motions of celestial bodies with great precision.
Unless a meteor hits the earth and carves out the moon.1 Or not. It is easily conceivable that a solar mass ejection would alter a meteor's trajectory enough that a few millennia later it would not hit earth. And I think it is obvious that processes in the sun's convection zone, and thus ejection events, are entirely non-linear and therefore prone to differ from "run to run" because of large-scale effects of quantum-randomness.
No moon. No tides. No tide pools. No advanced life. No humans. Oops.
1 Similar arguments can be made for larger-scale events like what gets hit by a relativistic jet depending on a quasar's precession etc., or how a super nova's explosion remains exactly look like (I suppose that's a fairly non-linear event, too). The latter would have significant large-scale consequences for the further development of a region in a galaxy.