Please read recent updates at the foot of the post
To get an invite to Google+ now simply ask a current Google+ user to share a post with you (the non Google+ user).
When you click on the Learn more about Google+ button in the email you receive, you should be taken to a page to sign-up for Google+.
We’ve just tried it and it indeed works.
It’s worth noting that invites are rate limited by hour. So if they can’t click through that first time, they can try again later with same link later.
There is in fact now no need to do the post trick mentioned below. Google+ users can now just share invites. If you’re already a logged in user, just refresh and you should see an invite option to the right of your stream.
Invitations are once again closed, and neither this workaround nor the invitation tools will work until they are once again opened up.