It should be mentioned that Game Activity now appears as Registered Games. If you're using an older version of Discord you should see Game Activity, but if Discord is updated you will not see that option. Another thing to consider is that the 'Game Activity' option is only available in the Discord application. If you're using a web browser, you will not see 'Game Activity.' If you're looking for this option, you must use the desktop application. You will see the Registered Games option (as shown above). But you will not see the Add it button, meaning you can't add a game to Discord using the web browser.