Are there parts of the brain which don't change over a lifetime? We know that during our lifetime the brain develops new neural connections. In addition, there is pruning of neural connections. These phenomena together are called neuroplasticity. 

Is there any research on part(s) of the brain that are fixed in terms of structure and other properties? In other words, are there regions in the brain where neuroplastic changes do not occur?
First off, 'neuroplasticity'is a broad term. Although structural changes are the most visible form of neuroplasticity, for example neurogenesis and axon outgrowth, generally subcellular changes are also taken into the definition of plasticity as well, for example the strengthening of synapses that occurs at the molecular level:


  Neuroplasticity occurs at different levels: structural plasticity (new neurons born, growth and shrinkage of neurons), functional synaptic plasticity (enhancing/diminishing the functional connection between neurons) and all the molecular and cellular mechanisms that accompany these changes


Secondly, the 'is there an exception to the rule' questions are notoriously hard to answer, as most scientific research reports pertain to the publishing of positive data (descriptions of neuroplastic changes in areas of the brain). 

To cite Aguire: 


  ...[A]s long as there are neurons, there is a potential for neuronal plasticity...


Hence, given the broad definition of neuroplasticity, I dare to state that it basically occurs in every brain region. In some more than others, however. The hot spots in the brain where neurogenesis takes place even in the adult brain, e.g. the hippocampus, are obviously the prime examples of high potential for gross plastic changes to occur.