How do I grow an invitation only site?

Assuming your site is something your users find worth inviting their friends, your best bet is to A/B test the hell out of your invitation process.

What percent of your user's send out invites? What can you do to improve this? What percent of sent invites lead the recepient to your site? What can you do to improve this? What percent of these convert to an account? What can you do to improve this? What percent of new accounts stay active? What can you do to improve this?

Is there anything you can offer your active members as reward for sending out invites? E.g. Dropbox offers additional space for each successful invite and that helped them grow.

Every site is different. What works for someone else might not work for you. That being said, your question is too broad and generalized.

I agree with the two posts about offering your users something and testing the percent of invitations sent out. I believe expanding your site through this model takes two things, some type of reward for bringing in new members even it is is marginal because it will make the user who is inviting people that much more likely to send out invitations and secondly, most obviously, having a great product. If you have a great product, something that people truly find useful and important, user's will without question want to tell their friends. People love to be the first to tell their friends about something useful and not only do they have that ability with an invitation model but they are in a position of power by being their "in" to the website. The key is creating a product so great that users will have to talk about it. Also depending on what type of site you are running, because you did not go into specifics, you have to examine whether the user base is better by having more users. Is this a social network? In a social network more users is obviously better for people because it allows them to connect with more people and more of their friends. The only value in a social network is by having people to interact with so the user base would be very motivated in adding people. The same could be said if this was a skill gaming site where having more people means more potential money to win meaning a higher incentive to invite people. The nature of your site will highly influence if people want to invite other people to it. You have to distinguish your's and if its not one that is made better by inviting more people then you have to focus on creating a great product and providing users with a reward for sending you business.