I like getting Growl notifications for incoming emails in Mail.app, but I'd prefer not to have to keep Mail.app running to see them. Is there a setting or third party tool for this?
If you're just using Gmail (like OP), then you can use the program Gmail Notifr. It lives in your menubar, and checks Gmail at a specified interval, it also supports Growl notifications.
It is NOT possible to get mail notifications without having the mail application active.
There is no way for notifications to know when you got new mail if there is no active connection to your mail server.
If you want, you can hide the mail app window (red dot).
But as always there are alternatives.
Assuming you use gmail and assuming you use Chrome browser than you can set it up to get your new eamil notification without using Apple Mail.
Desktop notifications in Chrome
Finaly there is another alternative.
If you care to do some set up and some scripting, you can use the Postfix.
Mac OSX comes with the
, which is a fully featured SMTP server.
There are plenty of instructions on the web how to do that.