In this question, I am defining neuroplasticity as being the creation of new connections between neurons. I'm aware that there is a high degree of neuroplasticity in the cortex and that new connections are created frequently in that brain area. However, in what areas of the brain are the connections constant or change rarely? I'm assuming that any brain area that controls muscles (such as the superior colliculus in the midbrain) are statically connected.
Subcortical structures can definitely show neuroplasticity in adults. Most likely, all brain structures can show plastic changes to some degree.
The question is rather broad, as there are many subcortical structures and neuroplasticity is age-dependent. I will therefore restrict my answer to three examples I dug up from the literature.
For example, in primates it has been shown that thalamic and brainstem areas projecting to the somatosensory cortex are plastic and change their projections to the cortex to aid in the rearrangement of the somatotopical map (Jones, 2000).
As a last example, subcortical plasticity has been shown after lesioning the spinal cord in monkeys and rats. By interrupting the ascending projections of mechanoreceptor afferents of the forelimb and the rest of the lower body the corresponding part of the somatotopic representation of primary somatosensory cortex becomes completely deafferented and unresponsive to tactile stimuli. If some of the afferents from the hand remain intact after such dorsal column lesions, the remaining afferents will extensively reactivate portions of somatosensory cortex formerly representing the hand over the course of a month. This functional reorganization has been linked to the sprouting of preserved afferents within the cuneate nucleus of the dorsal column–trigeminal complex in the medulla, also subcortical brain structure (Kaas et al., 2009).
In terms of structures that are resilient to plasticity, Aguire from USC argues that every brain structure has the potential for neural plasticity to occur. Basically, because the term plasticity is pretty much all-encompassing, and I quote:
[N]europlasticity is the ability of elements in the brain to show structural and functional changes in response to internal and external events. Neuroplasticity occurs at different levels: structural[ly], functional[ly] and [through] all the molecular and cellular mechanisms that accompany these changes. [...] At all times, these players are constantly changing [...].
Admittedly it is a popular-scientific and broad answer to this question, but I do agree with it.