A better solution is to require two invites to sign up. And then the service provider hands out double the number of invites. But now every new user has at least 2 existing contacts on the service - so their first experience is more than just saying "hi" to the only person they know there.
I think that would be impactful, but would still create barriers to entry. I'm not one to beg for invites, so I have a hard enough time casually finding one invite, let alone two. I'd imagine a good portion of invites would get stuck in purgatory waiting for a second, all while the potential user moves on to the next thing that grabs their curiosity.
What about one invite to create an account, a second to participate?