This is what I'd been doing:

1. Go to https://plus.google.com/circles

2. The first rectangle before the listing of your contacts should "Add a new person"

3. Enter email address of invitee and hit Enter

4. Enter a name to associate with above email address, select Circle to add into, and hit Save

I'm not entirely sure that this is the 'correct' way - I was just looking around for a place to enter arbitrary email addresses and this seemed like it. A couple of chaps I'd added this way got in, a couple more had to try a few times, and most of them see the "we've temporarily exceeded capacity" message.

Once you add people to circles, there's a bar across the bottom of window offering to explicitly invite those people to Google+.

Click to pop a dialog listing the people as tags or buttons, add more if you like, then invite.

If you leave and come back, the bar isn't there.

This is the same popup shown by the Invite button on your home page when it's there, just a different way of reaching it.

I've shared links with friends. Most of them got invites even though I didn't have any Invite link on the UI.

But it looks like this doesn't work anymore.