Is there a way of getting the output of a SHOW DATABASES or SHOW TABLES command to output to a text file, similar to how the SELECT ... INTO OUTFILE works?
The INTO OUTFILE produces a syntax error when used on a SHOW command. I am open to external commands if there is a way to produce this using a cmdline tool such as mysqldump.
The best way to output this would be to pipe the data to a file. For instance:
mysql -u root -e "SHOW DATABASES" > my_outfile.txt
@Velko's answer is a good answer but only if you can access the server file system. If the server is on a different system than the client, piping will be the only way to get the file on the local client system.