Everything including my usual desktop environment keeps working. I have /home on a separate partition though.

Most of the time I only notice that it filled up again when pacman -Syu fails because it can't save anything.

It probably really depends on your setup and how stuff is configured. It appears the various systemd tools keep working just fine though, otherwise my system wouldn't even boot.

If I could make a wish, I'd want my system to just buffer writes (transparently) to RAM once the SSD filled up. Then fire some events that can be intercepted by a GUI, so the user could be informed that he'll lose data if he shuts the system down now. Then as a last resort, abort any shutdown the user initiated and drop him into a shell so he can fix the mess and allow the system to save his data.