Would our body adapt to changes that are not in our DNA?
It will be hard to discuss for the specific example you consider, mainly because it is unclear how would you treatment affect the person's height.

That being said, yes, our bodies can change in function of the environment. This is called phenotypic plasticity


  Phenotypic plasticity refers to some of the changes in an organism's behavior, morphology and physiology in response to a unique environment.


Here the term environment does not only refer to outside temperature or other obvious environmental factor. For example, one can ask whether the size of a vein can react plastically to the blood flow. Here, blood flow is the environmental factor toward which, the plastic response is mediated.

Note that some plastic responses are adaptive and some are not. Note also that there are a lot of related terms (developmental flexibility, acclimation, polyphenism, developmental selection, ...) in the field of plasticity and different authors sometimes use these terms with slightly different definitions which can make everything a bit confusing.