The nature of information, and the definition of simple versus complex, is surprisingly tricky. Scientific definitions, while in a very general way similar to common usage of these words, differ in full technical detail, so that one's intuition must be trusted with extreme caution. For example, those gorgeous fractal patterns one can see everywhere nowadays, are generated by very simple mathematical formulas. Similarly, a pretty simple mathematical algorithm, when run for a long time, generates the value of pi to as many decimal places as one could want. Pi is an irrational, and moreover a transcendental number, which never repeats and has no obvious internal "pattern", yet can be generated by a simple calculation. A truly random series of numbers has no internal pattern, and by definition therefore cannot be generated by any simpler formula or calculation, it can only be described by writing the number itself to as many digits as there are. But for a given large number, with no obvious internal pattern, it's very hard to be sure that it cannot in fact be generated by a simple algorithm (i.e. set of calculations). If I understand correctly, it can be proved that it is impossible to prove that there is no possible algorithm to generate a given output. So, short of actually finding a simple algorithm to generate a number (like pi), it's hard to know that a given number is truly truly random - maybe someone smart enough will figure out the algorithm in the future.

So much for pure math. The human genome has approximately 1.5 billion "unique" bits of information (i.e. GATCs), which can be thought of for some purposes as a code or "number" containing data, but it's very hard to known how much actual information is contained. By comparing our genome to those of other animals, some estimates are that only about 1% of our unique DNA sequence is highly conserved, though other estimates are much higher - it depends to some extent on which other animals are used for the comparison. At this point in time we don't know the answer.

So much for the genome in the abstract. There's no question regarding the tremendous complexity of the human brain in terms of the absolute number of neurons (cells) and synapses (links). But it's very unclear how much genomic complexity is required to create the structural complexity. Maybe the underlying algorithm is very simple, and is simply repeated a large number of times, like fractals, only with lots of feedback from the environment so that no two human brains end up working the same way. I like to keep in mind that many types of plants are biochemically much more "complex" than we are, in that they can carry out many more types of chemical reactions and have more different types of small molecules (like glucose, ATP, acetyl-CoA, etc) than we do. It is somewhat animal-o-centric to hold that complex 3-dimension bodies with lots of different tissue types are "more complex" than complex biochemical networks. Plants might feel that they are actually more advanced, and better adapted than we are too. If the amount of oxygen in the atmosphere dropped from 21% to say 15%, I imagine most living animals, or at least mammals, would quickly go extinct, whereas many plants could probably adapt quite well. Superior adaptation to a specific environment does not necessarily equate with overall success under environmental change. Let's see who's left standing 100-200 years from now, after the global average temperature goes up 5 degrees C. I predict there will be a lot fewer humans, but probably about the same number of cats!