I actually got my invite from someone who works at Google, but I keep getting the "We've temporarily exceeded our capacity. Please try again soon." message when I try to join.
My friend says keep trying, but I might just wait until they open up to more people.
I don't think its a loophole, It must be intentional.
Second way which is popular now, you can add the gmail ids in your circle and share a comment to the circle. This will send a invite to everyone in the circle :)
On your stream page just type a message in the share something box "Hi." for example. Then in the sharing selection box put in their email address. They'll get a message with a link to see what you shared with them and for whatever reason that seems to have a much higher success rate though it can still take several attempts at clicking the link in the email before they get the sign up instead of the we're full message.
Also, if you are sharing with a circle you have to make sure you check the little box during circle selection that says "Also email X people not yet using Google+" or they won't get the email.
About 90% of the people that I've sent invites to are signed up now.
This was how I got invited (out of the blue invite advertising some site) and how I invited over 20 people over a 24 hour period (tagging them in a photo/status update or sharing a status update to a circle/email address). They didn't get any "Exceeded Capacity" messages although the real invite UI still doesn't work. And it was after I heard on HN/Twitter that "the loophole has been closed". Maybe some cliques are more welcome than others.
You don't need the invitation via email to get into Google+, just tell your friend to invite you (tagging or posting, whatever). Go to http://plus.google.com. Log out of your Google Account. Log in back. Voila!
If you make a circle with email addresses of people who don't have Google+, then make a post to that circle, they will get invites. The catch is, that even if you have an invite, Google+ will still just put you in line instead of letting you sign up immediately.