Preface. I DO NOT HAVE ANY GMAIL INVITATIONS. I have given them all to friends, co-workers, and one random IRC acquaintance.
We know that Google's parceling out of Gmail invitations was a really smart marketing move. The invite system made the service seem that much more exclusive, and thus that much more desireable. However, I'm betting it wasn't their primary motivation. I suspect there was a far more important reason to launch the service this way: stability.
Here's a possible scenario.
Google finish the beta program for Gmail, and open subscriptions to the public. Over the next few days, millions of people subscribe and explore the interface. Many start pumping their mail archives into the service. This creates a load-spike orders of magnitude higher than the service would normally have to maintain.
At the same time, some Google engineer starts discovering that a number of parts of Gmail that they thought would scale linearly, don't.
Over the next month Gmail is down more often than it is up, none of the programmers are getting any sleep because they're too busy putting out fires, and some sub-editor at the New York Times decides "Gfail?" would make a good headline for the post-mortem, which really isn't good publicity for a company looking towards its IPO.
Invite codes are a really smart way to control growth. After a few weeks, anyone "in the know" enough to really want a Gmail account, can have one. At the same time, Google can control the service's rate of growth. There's no explosive rush, and if anyone notices something isn't scaling as well as it should, they can throttle back the issue of new invitations until the problem is solved, without too many sleepless nights.
Unfortunately, it's the sort of tactic that only works if you're as desireable a service as Google: giving Bob three invitations only helps if Bob knows three people who want to sign up. If you're just a startup with a good idea, an invitation program would just kill any chance of your service getting traction.
I wouldn't be surprised, though, if I saw future big-name MMORPGs opening their doors post-beta with an invitation program.