After several months Google Analytics has opened their doors again to new subscribers on invitation basis. They are inviting based on email addresses submitted.

From their web site:

We've added enough capacity that we are re-opening signups on an invitation
basis. Many of our users who previously submitted their email address to us
will be receiving an invite shortly.

As you may know, we experienced extraordinarily strong demand shortly after
we launched last November. To ensure a high quality user experience for
existing customers, we had to temporarily disable new signups. We have been
working diligently to add more capacity that we can both enhance performance
for existing users and re-open signups to a limited number of new users.

Since we launched, we've improved the freshness of report data so that our
existing customers can see reports that are as recent as 3 hours ago. We've
also re-enabled the addition of new profiles so that all of our existing
customers can track additional sites using Google Analytics.

Currently, we don't have enough capacity yet to re-open signups for everyone
who wants an account. Not everyone who previously sent us an email address
received an invite. We will continue to send out additional invites as we
add more capacity.

Thanks for your patience and for your interest in Google Analytics.

-The Google Analytics Team.