Can we artificially increase human growth using HGH?
You asked


  ... is it theoretically possible for someone (like a "fully grown" adult) to artificially grow beyond their genetic potential by providing them with regular doses of HGH? 


For an adult, the answer is no. Human growth means skeletal lateral growth, which is indeed mediated by human growth hormone. But during normal human development bones ossify in the process, and in human adults the bones are completely ossified and can't grow in length anymore. 

From the cancer.gov site on Bone development and growth:


  When cartilage growth ceases, usually in the early twenties, the epiphyseal plate completely ossifies so that only a thin epiphyseal line remains and the bones can no longer grow in length


Further reading: NEJM Osteogenesis: The Development of Bones

This point is usually reached around the early twenties in humans. The case mentioned in your question is different because in this case, the human growth hormone was "given" before this process had finished and thus the bones grew longer. He was 22 when he died, so we will never know if he would have finished growing in his mid- to late twenties. 

Giving adults (that don't suffer from an HGH deficit) more human growth hormone has been done and does not result in lateral growth, but it does result in increase in lean body mass in both young adults and the elderly