I know bots are getting smarter and they can pass through captcha tools. You can add a math problem that needs to be solved and it is unique to your website, but I'm not sure how effective that will be. It is also bad user experience. I still recommend using services such as Cloudflair. They identify bots and monitor their traffic in a distributed fashion across the web. It also improves the speed of your site a bit.
I stumbled across this a couple weeks ago: https://spamicity.info/. It's a spam detection system specifically for social settings. It looks interesting, but its just an API. Someone would have to build an implementation for Anahita.
@rmundel our members have to send a request and we create the accounts manually. Usually 1 in 5 make it through, because I care about building a community of people who care.
Another good approach is charging a very low membership fee like 1$/year to make sure only those with a credit card can get it, but it isn't an approach that will work for you if you are trying to build a user-base for a free service.
I'd give Cloudflare a try in your case. Bots mimic the behaviour of popular browsers so there are no simple ways to differentiate them from people. I'm glad to know that you haven't had any SPAM registration so far.