What is the name of this strange white caterpillar?
It is hard to be sure without clearer pictures, but that looks like something in the genus Eriocampa and it could be a Butternut Woollyworm (Eriocampa juglandis).
Picture taken in Kentucky from BugGuide.net © 2015 by dan40165:

These are the larvae of a sawfly (so not technically a caterpillar) and are about 1 inch long.
You can learn more about this particular species here. To narrow down the exact species it would be helpful to edit more information into your post including the types of plants in the area.