We do at ArchiveBox.io. We've had some minor issues with multiple processes trying to do long write transactions at once (SQLite can only handle one writer at a time), but other than that, no regrets.
I planned to support PostgreSQL initially but as time goes on I yearn for it less and less. It's greatly simplified the web stack by not needing an init system to spin up a DB/redis/etc, and it allows people to get started with a simple `pip install archivebox`, which is one of the main selling points of the project.