Use an app like instazoom. These apps allow you to view and download the full size Instagram profile picture.
Go to the Instagram website and open the profile whose profile picture you want to enlarge. Then right-click on the profile picture and select "Open picture in new tab". The profile picture will now open in full size in a new tab.
Use a browser like Chrome or Firefox that supports page source view. Open the Instagram website and open the profile whose profile picture you want to enlarge. Then right-click an empty spot on the page and select "View Page Source". Scroll to the bottom of the page source code and look for the profile picture tag. Copy the link to the profile picture tag and paste it into a new tab. The profile picture will now open in full size in a new tab.