I am aware of five categories of phenomena
     that begin with simple conditions
     and evolve to a state of complexity.

The first of these is combinatorial systems.
A combinatorial system incorporates
     a set of symbols, elements, or rules that model reality.
These symbolic sets are familiar to us as alphabets,
     numbers, musical scales, DNA, the Periodic Table, etc.
Just a few symbols or elements are combined to form
     highly complex systems such as writing, mathematics,
     music, life forms, physical objects.

The second class of phenomena is cellular automata.*
Cellular automata are generated with a computer program.
Typically, a computer programmer generates graphics
     based on a simple set of rules.
The rules specify what the program should do at each step.
For example, the program may first display a row of black and white squares,
     selected at random.
The program then iterates thousands of rows of black and white squares,
     applying a simple rule set to each new row.
The rules may be simply that if a square in the same position in the previous row
     is black, then make the current square white, and do the opposite for a white
Interestingly, a simple rule set can turn out to generate fantastically complex 
     schemes over time, ones capable of modeling nature.

Another phenomenon is symbiosis.
The first organisms were simple systems: bacteria and viruses.
Biologist Lynn Margulis has demonstrated
     that under extreme environmental stress,
     one type of bacteria would invade another.
Sometimes the invader would fail to kill the host.
Likewise the host would be unable to repel the invader.
The result was that a new kind of relationship was formed;
     an organism would emerge more complex than either of the two alone,
     better able to compete, survive, and reproduce.
In addition, external relationships between two species, such as flowering 
     plants and insects, or trees and fungi, have evolved complex systems.
Symbiosis is a continually evolving biological process
     that is responsible for increasingly complex morphology
     throughout nature.

A fourth class of phenomena is physiosynthesis.+
The physical (as opposed to biological) interaction
     of two or more dissimilar structures,
     which combine to form complex systems,
     occurs at various orders of magnitude and scale.
String theory suggests that there are tiny vibrational patterns
     that constitute the smallest known forms of energy.
These strings oscillate in various ways,
     producing fundamental particles of energy and matter,
     such as photons, electrons, and quarks.
Electrons and quarks form the structural basis of an atom.
Atoms interact with one another to produce molecules.
Atoms also interact with other atoms to form stars.
Different molecules bond with one another,
     creating a variety of structures and forms
     ranging from a water molecule, to DNA.
Large molecules combine to form cells, plants, animals, ecosystems.
Reactions among molecules produce larger structures such as oceans, 
     mountains, continents, planetary systems, and even giant molecular  
     clouds that may act as nurseries for newborn stars.
At an even larger scale, there are examples of astronomical bodies such as
     galaxies that violently interact, through collisions, producing new, more
     complex, galactic systems.

Finally, atoms and molecules, in conjunction with biological evolution, have
     combined to create the diversity of life on Earth, including the most complex  
     system in the known universe, a human brain.
Human technology, in concert with the developing brain, has evolved
     from simple tools invented millions of years ago, to complex technologies 
     created in the 21st century.

* Stephen Wolfram, A New Kind of Science
+ an idea presented in the Nature and Inquiry group by Ron Wallace; the name
        physiosynthesis is my own