What would most likely happen to a person who is not getting enough essential amino acids in his or her diet?

If you don't ingest enough amino acids you would have difficulties synthesizing protein. Basically, there are 20 of them in the genetic code and they build into everything because they are combined into proteins. They are the second most found molecule in our body after water, and you need a lot of them. If you don't get them your body won't have enough protein and you can die if it becomes too extreme.