How to disable the button next to Activities in GNOME?

Solution 1:

The button next to "Activities" is called "App Menu".

App Menu can be designed to contain not just "close" option, but other options related to the corresponding application, e.g. Preferences/Settings, Help, About etc. But almost all the third-party applications (if not all) are not yet designed to contain any options other than close. If you run native GNOME applications like Files (aka Nautilus), Text Editor (aka Gedit) etc. you'll find there are options other than close.

To remove App Menu from the top-bar you may use (GNOME) Tweaks. First install it by running

sudo apt install gnome-tweak-tool

Then launch Tweaks, go to "Top Bar" section and toggle off the "Application Menu" option.