In a weird combo of events today, Amazon has discontinued their "Free App of the Day" promotion and simultaneously opened their new Amazon Underground app which offers a reported $10,000 in free apps. Not only are the apps free but in app purchases for those apps are also reportedly free.
Getting the app takes a couple extra steps though.
To start, head to Amazon's Underground site from your browser, enter your phone number or email address and hit send. This will send a link to your phone that will allow you to download the app. If you navigate to the site from your phone, you will be able to click the download link without entering the information. You will need to enable the ability to install apps from Unknown Sources in your phones settings. Then navigate to the downloaded file, tap on it, and install it.