I was unable to find a truly "minimal writing app", so I made one.
It has quite basic functionality, no support for rich-text editing or markdown, no auto-complete, only very basic customization capabilities. I wanted the ability to go into the fullscreen mode with a completely blank space in front of me and a single pulsating caret, and that's what I created. I also wanted something that I can load extremely quickly before I lose my train of thought when the inspiration strikes.
No user logins or anything, in fact, the app is written in plain HTML/Javascript, so you can easily see and edit the code.
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