The depth of the sub-directory should not matter.
If you are writing to files to the directory based upon a user-run script, the folder permissions may need to be set at 666 or 777. If you are just updating a single file, you can just change the permissions for the file.
An alternative is to change the owner of the file to nobody. You should be able to keep your current permissions and it will add a better level of security. Depending upon your hosting arrangement, you may or may not have the ability to change owners.