Any way to create desktop app/game?

Question

I have an idea to make a simple game on the desktop, so that it is just always open and right on the desktop.
Either without an application window, or with a transparent background that will pass inputs on desktop in fullscreen. I tried setting the alpha channel to 0 on the environment background, but it does not become transparent.
For example, a clicker, so that you can play it while on the desktop
I hope I have explained clearly.

Answer 1

Go into Project Settings > Display > Window and set:

Size > Borderless → On
Per Pixel Transparency > Allowed → On
Per Pixel Transparency > Enabled → On
 
If you want the splash screen invisible too, set:

Project Settings > Application > Boot Splash:
Bg Color → # 00000000   (or A = 0)

Also you can set the splash Godot logo invisible by setting the Image to a 1PX transparent image, like this one:
https://i.imgur.com/QNIf7W2.png (right-click > Save link as...)
 
Further Details as per the followup:

In the WorldEnvironment (or default_env.tres) set:
Ambient Light > Color → # 00000000

And finally, it's still necessary to enable transparency at runtime with the script:

func ready():
    get_tree().get_root().set_transparent_background(true)

Hope it helps!