In answering your question I am assuming you are referring to brain growth and development from birth onward, not during intrauterine fetal life.
The growth and development of the human brain does indeed double with time, however there are marked differences in the doubling time among the myriad functions of the brain compared to the physical growth (enlargement) of the brain.
The infantile brain is said to be “plastic”. This is positive in that the plastic brain has a greater capacity for learning and adaptation. On the negative side, however, the plastic brain is susceptible to developmental defects in non-nurturing or hostile environments.
In human brain growth and development, genetics are important for the formation of proper brain structure and wiring. Environmental effects on brain development determine the brain’s ability to properly function and adapt.
In terms of doubling times for the brain, here are some facts:
Brain Size: Newborn = ¼ adult size, Age 3 = 80%, age 5 = 90%.
Number of neurons: 100 billion. Peak occurs before birth.
Speed of Neuroprocessing: slow in infancy, maximum at age 15, gradual decline thereafter.
Language: development of language skill begins at 5 years and is essentially completed by puberty.
Other skills: there probably are not specific critical developmental periods for each and every brain function. Most brain functions depend on correct development and function of sensory organs, such as vision for hand-eye coordination and perception.
Important co-factors that affect growth and development of the brain are nutrition, stimulation (sensory, auditory and visual), nurturing and exposure to language.
The end result of brain development is the acquisition of normal abilities in perception, memory, thinking and feeling (emotions).
Please see the reference for a detailed description and explanation of the complexities of human brain growth and development.