Instazoom is an app for Android devices that allows you to zoom in and download Instagram profile pictures. You can simply enter the username and the app will show you the profile picture in high quality. Remember to respect the privacy of other users.
There are also various online tools that you can use to enlarge Instagram profile pictures. These usually work by entering the link to the user's profile. An example of such a service is “Online Profile Picture Enlarger”. Again, caution is required to protect the privacy of others.
Instazoom is another app for iOS devices that allows you to enlarge Instagram profile pictures. You can search for users and zoom their profile pictures without being notified. Be sure to follow Instagram's usage guidelines.