Which type of protein penatrates the plasma membrane but does not go all the way through it?

The answer should be membrane proteins.

In terms of the membrane proteins, the classes of proteins that penetrate the plasma membrane but does not go through it, completely would simply be integral proteins, these proteins play fundamental role in transporting relatively large polar substances and other molecules. Transmembrane proteins are a specific kind of integral proteins that span or go through the plasma membrane, completely.