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Watch this video describing tissue elasticity in Marfan patients.
The human has many organs that utilize elastic characteristics of tissue, including:
Muscles
Cardiovascular system (heart, valves, blood vessels)
Eyes, especially the flesh that controls the stretching and contracting the lens
Bones, yes bones are pretty solid but they have some flexibility i.e. elasticity
Teeth, palate, jaw as these are actually part of the bone system
Lungs
For more info, see marfan.org organ Key Features here.