The Kolmogorov complexity of an object is the length of the minimal description of that object. In terms of the brain, it would correspond to the length of the smallest computer program that could reproduce the brain. It could also be thought of as the amount of information necessary to model the brain. Computing the Kolmogorov complexity is not possible since it is an undecidable problem but we can estimate it. If we presume that molecular biology is computable then one estimate of the Kolmogorov complexity of the brain is given by the length of the genome, which is 3 billion base pairs long or 6 billion bits. To be conservative, we could also include the genome of the mother and baby, which implies 12 billion bits. This corresponds to less than two billion bytes and easily fits on a DVD. Hence in principle, we could potentially grow a brain with less than 12 billion bits of information and this is probably an upper bound.
However, this would not imply that we could describe a brain with this amount of information since it ignores the modifications due to external inputs. For example, the visual system cannot fully develop if the brain does not receive visual inputs. So we also need to estimate how much input the brain receives during development. The amount of information available in the external world is immense so it is safe to assume that the amount received is limited by the brain and not the source. However, there is no way to estimate this in a principled way since we don't know how the brain actually works. Depending on what you assume to be the neural code (see previous post), you could end up with a very wide range of answers. Nonetheless, let's suppose that it is a rate code with a window of 10 ms. Generally, neurons fire at rates less than 100 Hz so this corresponds to the presence or absence of a spike in a 10 ms window. This corresponds to 100 bits per neuron per second. The brain has about 10^11 neurons so the maximum amount of information that could be input to the brain is 10^13 bits per second. There are over 30 million seconds in a year, so that is a lot of information and can easily dwarf the genomic contribution.
However, this does lead us to a potential means to quantify the influence of genes versus environment on intelligence and behaviour debate. If the complexity of the brain is less than 12 billion bits then we are basically genetically determined. If it is greater, then we are mostly shaped by the environment. So what do you think?