It seems to be everywhere. In a previous question about chaos I mentioned that even in our solar system the orbits of the planets have been found to be chaotic. Not knowing the location of Mars to an accuracy of 1 millimeter was found to have a effect on the orbits of the other planets after millions of years of calculation to determine their predicted positions. There are many other situations in which complex 'non-linear' equations have to be used to predict the internal structure of stars, the gravitational dynamics of galaxies, or the evolution of structure in the universe. All of these systems display a certain sensitivity to the exact input values you use for position and velocity. Chaotic motion in many of these examples does not lead to anything qualitatively new, just simply a slight elaboration of the possible final configurations. In other words, astronomers modeling the convection patterns inside a star are not interested in the exact, to the centimeter, location of the convection cells, they just want to know in a statistical sense how they transmit energy from one level to another inside a star. In cosmology, although chaos can also be important, astronomers are often only interested in matching the sky we see against a predicted sky with the same statistical appearance, not exact positioning of galaxies to a fraction of a light year!
The significance of chaos has to do with what aspects of a system you are trying to model. For the solar system, small differences over millions of years are important; for other studies they are almost completely irrelevant, at least given the kinds of questions we are asking today! For more about this, have a look at the related questions:
Question 190 Question 355 Question 364