Register user without email address

When creating a user in WordPress an email address is required for each user. The website I’m working on requires that users remain anonymous (it’s part of a cancer research study) and email is an identifying piece of data. Users will be self-registered and I am using Gravity Forms. Ultimately, the only information they will provide is a unique username and password.

Am I going to have to provide a fake email address for each account, or is there a way to leave this field blank?

What I want to do is very similar to this: https://wordpress.stackexchange.com/questions/22754/user-without-email

Except the code here broke my site. Any info/insight is appreciated.