How can I maximize the editor pane in IntelliJ IDEA?

The closest thing would be to hide all tool windows by invoking the Hide All Tool Windows action. The shortcut for that is Ctrl + Shift + F12 (Default keymap).

This will hide all tool windows, effectively maximizing the editor window (though not full screen). The IntelliJ menu bar, toolbar, breadcrumb and tab bar will still be visible.

I'm using IntelliJ 11.1.2 on Kubuntu 12.04 LTS with the Default keymap.

14.0.3 on MacOS X

It's Cmd + Shift + F12 in IntelliJ IDEA 14.0.3 on MacOS X.

UPDATE on 2015-03-24: IntelliJ IDEA 14.1 now has support for Distraction Free Mode. You can invoke it by clicking View > Enter Distraction Free Mode. In this mode, IntelliJ hides everything but the menu. For more details, follow their video detailing the new feature: https://www.youtube.com/watch?v=uVwE8MFgYig.

If you want to maximize a pane , select that pane (by clicking inside or on its title bar) and then use the shortcut "Ctrl+Shift+Quotes"

You can double click anywhere on the tab title bar of the editor pane.