Examples of events that are unpredictable as a matter of principle? Are there examples of phenomena/events/states of affairs (henceforth "events") whose outcomes are thought to be unpredictable, even in principle?  
I.e. are there events such that if one had access to all the relevant information, and knowledge of all the relevant principles and laws of nature, whose outcomes could not be accurately predicted as a matter of principle (as opposed to in practice / with current technology / with current incomplete understanding of laws of nature etc.).
I'm looking for a sort of event A where, as a thought experiment, we imagine going back in time to prior to the event A and holding all the conditions leading up to the event A exactly identical, we could still expect some different outcome B the second time around.
I appreciate that the answer here may hinge not only on facts, but also on an interpretive framework for those facts - if so I'd like to hear about a relatively mainstream interpretation.
Edit for clarity: I'm interested in any sort of event at all, radioactive decay, electrons moving down orbitals, etc., etc..
 A: 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.
A: 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.
A: 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.
A: Well, in the realm of quantum mechanics, much is unpredictable. For example, electrons in an atom exist in a cloud around the nucleus, and due to the Uncertainty Principle, you can never accurately know the position and the momentum of an electron, as a matter of principle. Therefore it is impossible to predict where exactly you will find that electron until you measure it. It's intrinsically random. This property disappears on larger scales, so I'm not sure if there are any other examples of completely unpredictable behavior. I hope a quantum mechanical explanation will suffice.
A: Yes, of course, as stated by the answer from @Schwern, a quantum measurement is unpredictable (unless you have specifically prepared a state for the microscopic system, and there's been no interaction with it, in which case you recover the same state). I mention in this answer some  other interesting information, and then imperfections. 
Also, just to be clear, I am not commenting on the interpretation of quantum theory as either waveform collapse, or the Bohm pilot theory, or any others. Any interpretation has to agree with the observations and measurements we make in quantum theory, so local/not local, realist/not realist, are different issues than those discussed below. Whatever the interpretation (i.e., hidden variables or not), it is clear that quantum theory leads to random results in measurements. 
For the radioactive case the emission of the radioactive particles happens purely randomly. The statistical distribution is a random Poisson process. 
It's been possible in the last 15-20 years to measure the properties of individual photons, atoms, electrons and ions, and probably more. The tunneling electron microscope (1981) uses individual electrons to image the surface of a solid, and 'sees' the individual atoms. See https://en.m.wikipedia.org/wiki/Scanning_tunneling_microscope
Individual atoms and ions have been trapped and measured. The most intuitively clear case is observing quantum systems where the state can only be two, such as the polarization of a photon (if circular, it can only be right or left, i.e. only two states), or an electron which along any one axis can only have spin up or down, or the state of a 2-state trapped ion (rest state and one fairly stable excited state). These are done in quantum computation labs where some devices are used to use a photon or an ion and measure their states. If they are not prepared in a specific state, you will measure a random polarization or state or spin. 
So the question is then what is random, when you prepare them in a specific state? Consider the electron with a spin up in the z axis. If you now try to measure the spin in the x or y axis, for that electron you carefully prepared, it will come up randomly up and down. Just like position and momentum can't be defined or measured simultaneously exactly, spin in the three directions also cannot be simultaneously defined or measured. If one is measured, the other will be totally random. And there is no way you (or nature) can prepare a microscopic system so you can measure exactly all the values. 
You can also prepare a two state system in a superposition of the two states, say a superposition of up and down. When you measure it'll be 50% up and 50% down, completely random. 
All of that is in principle, simply from quantum mechanics, and no matter what measurement apparatus you use. See the wiki article on quantum randomness at https://en.m.wikipedia.org/wiki/Quantum_indeterminacy. 
Please note that many other systems we would think are random, can often be analyzed with a lot of data and determined as not purely random. That is true for chaotic systems, which are purely deterministic but are exponentially sensitive to initial conditions. Also often true for classical randomness, like flipping coins. Very minor and undetectable asymmetries in the manufacturing of the coin can after many runs be found to favor just a tiny bit heads over tails or the reverse, if you are patient enough, i.e., practical imperfections can often eliminate perfect randomness. But all of these are practical and classical factors. Random number generators ideally generate perfectly random sequences, but over a long time possibly the imperfections of the hardware can allow you to see some non-randomness. In a more complicated way you may prepare a quantum state, but there also can be imperfections. Thing is we've been able to create those machines and conditions in the lab well enough that we can now generate enough randomness with various processes where it could take a large number of universe ages to catch the randomness. So, in effect, and for practical purposes, it is random enough. Quantum theory seems to go beyond that, but of course in the end we build the systems, and our imprint is on them. You just sometimes would have to watch it for more time than will ever exist to see the non-randomness.  
A: Quantum uncertainty and sensitive dependence on initial conditions interact. 
I recall that snooker ball collisions amplify unknowns to the extent that given only Heisenberg uncertainty in the initial conditions, one cannot predict the outcome of 15 successive collisions.
Hence macroscopic systems as well as microscopic systems are inherently unpredictable given sufficient time evolution.
A: According to quantum mechanics, the outcome of almost any observation is inherently unpredictable.
More precisely:  Given any system (e.g. any particle or any collection of particles), in any state $S$, and given any observation whose eigenstates do not include $S$, the outcome of that observation is inherently unpredictable.  Moreover, taken $S$ as given, almost no observations (in the sense of measure theory or in the sense of Baire category, or in any other reasonable sense) include $S$ as an eigenstate.
