According to the following quotes I don't think you need worry about running out of space.

The Franklin Institute's Center for Innovation in Science Learning says: "The human brain is able to continually adapt and rewire itself. Even in old age, it can grow new neurons. Severe mental decline is usually caused by disease, whereas most age-related losses in memory or motor skills simply result from inactivity and a lack of mental exercise and stimulation."

James Watson, co-discoverer of the physical structure of DNA said ." The brain is the most complex thing we have yet discovered in our universe."

"A book by neuroscientist Gerald Edelman explains that a section of the brain the size of a match head "contains about a billion connections that can combine in ways which can only be described as hyperastronomical—on the order of ten followed by millions of zeros."