Back in the day, invitations to Gmail were so scarce and popular that people would pay for them. Today, the Gmail invite window is just a nuisance. As it turns out, if you wish to be rid of said framed box, simply give away all your invites. Personally, I just invited myself 100 times over; makes one feel all welcome.
Update: Alas, things aren’t so simple. Within a week you’re given 50 new invites. It seems you need the Better Gmail 2 Firefox extension to truly hide the box.