Given the right data being correctly collected, and analyzed using sound predictive models, what can be predicted, and what can't be predicted no matter what?
I believe that I have an answer to this question. All systems and processes that rely on some energy source can be predicted, and the other way around. Note that energy is sometimes called entropy by mathematicians.
Energy brings structure to systems. Without energy, we are faced with chaos and randomness. Structured systems can be predicted. However, systems powered by high energy sources are currently difficult to predict. Likewise, systems powered by low energy sources are equally difficult to predict, if one uses predicting modeling and data capture techniques of this century. It will be easier in 50 years.
Let me illustrate my theory (at this stage, it's more a personal opinion than a theory) with a few examples:
Weather predictions: Weather is powered by high energy levels (getting higher if global warming is true), making it harder to predict locally or short-term. Global, long-term predictions might be easier if performed by data-savvy and domain expert data scientists. This is typical of high energy systems: local predictions are harder to make, compared with global predictions.
Earthquakes predictions: Earthquakes are powered by high energy. Prediction is very difficult because, as of today, no one is able to collect the right data: history of stress and forces acting in faults. When this data becomes available, Earthquake prediction will become easier.
Car traffic patterns: Relatively easy to predict in real-time; the energy source is collective human brain power used to optimize individual car trips.
Stock trading: Significant collective human brain power is used to beat the market, making it harder to predict (in real-time) than car traffic patterns. Such systems belong to a category of processes commonly known as arbitraging.
Digits of SQRT(1/2): Are these digits random? If yes, it would mean that there is no energy force behind it. They look extraordinary random, which means that they can't be predicted, or the energy behind it is so powerful that it's hard to predict. Some people will claim that there is extraordinary energy behind it - they call it God - and some other people claim that these digits can easily be computed with a rudimentary 3-line algorithm, confirming that indeed, there's hidden energy behind it.
Structure in particles at zero temperature: Very stable and predictable structures (crystals) can be found at apparently very low levels of energy, in deep space. These atoms are actually subject to extreme nuclear forces and energy levels that bind protons and electrons together, as well as interactions with other atoms. This energy, when properly leveraged, can produce atomic bombs.