Does Deterministic Chaos ensure that there are certain phenomenon which will never be predictable?

Deterministic Chaos basically states that for a system to be considered chaotic, the system must be exquisitely sensitive to initial conditions. What this means is that even the slightest change in the initial conditions of a chaotic system can result in tremendous differences in the system’s development.

This is also known as the Butterfly Effect. This was named as such because a famous meteorologist found that even the slightest effects, such as a butterfly flapping its wings in one part of the world can have adverse effects on the weather in another part of the world. For this reason among others weather is considered to be a chaotic system. There are many other examples of chaotic systems as well.

Chaotic systems are extremely difficult to predict using linear means because linearity basically means that effects are proportional to causes. It has been found that for many chaotic systems trying to use linear means to predict their outcomes is very difficult. This is so because there are too many minor variables, both observable and unobservable, that will be a major factor in any chaotic system’s outcome.

I’ll admit that almost nothing is entirely linear, and even the basic concept of getting a baseball to go twice as far by hitting it twice as hard would likely be interfered with by some sort of unobserved conditions, but some phenomenon are clearly more linear than others regardless. My question here is what I wrote in the header: are there any phenomenon which exists that likely will never be reasonably predictable by using linear scientific methods that you can think of? Examples may include weather, psychology, physics or anything else for that matter.