Is there any free (as in beer and as in freedom) text editor that saves everything I write into it without even needing to name the file? That said, opening it again shows me the text I left.
Try Sublime Text 2. Make sure the "hot_exit" option is turned on. (It is by default.)
CudaText is free, cross-platform, it has this feature (since last version).
"ui_reopen_session": true (default), then on closing you see dialog "Save tabs?". In dialog press "Keep session" and unnamed buffers save to session file, and auto restore later.
Notepad++ is a great free editor which has this feature (and many more). https://notepad-plus-plus.org/