Sorry for a dumb question but I can’t figure out how to delete a database in DT. The intuitive way - I would have thought - is to click “Delete” when the database is highlighted in the list of open databases, but that doesn’t work because the “Delete” button is greyed out. Right-clicking also doesn’t offer the option of deleting the database.
You can’t delete a database from the filesystem from within DEVONthink.
If you want to permanently get rid of a database, close the database. Then, File > Open Recent > Clear Menu. With Finder, navigate to the database in the filesystem (it has the extension .dtBase2) and delete it.
Have a backup first!! You might accidently delete something valuable. 
I think that things must have changed since that original reply. If you right click an open database and choose ‘Delete Database’ and confirm the deletion it will certainly delete the database from the disk. Just thought it was worth pointing that out.
Certainly and thanks for the update. 