I always seem to miss it :(
It's the one in the top right with the UFO and antenna and it sometimes shoots lasers.
The actual gif is here: https://cdn.sstatic.net/Sites/gaming/img/header-foreground-image03.gif?v=330ce6fa0066.
Here it is uploaded to Imgur for your viewing pleasure:
It only animates once. After that, you need to refresh the page, or navigate to another page within Arqade.
Fun fact, it use to be the "Ask Question" button, and would animate when you hovered over it each time. But there was a site redesign that changed it to what it is now.