Recaptcha is broken in Joomla
Google changed Recaptcha API and Joomla plugin doesn't display Recaptcha :(
Edit /plugins/captcha/recaptcha/recaptcha.php:
1. Change
const RECAPTCHA_API_SERVER = "http://api.recaptcha.net";
to
const RECAPTCHA_API_SERVER = "http://www.google.com/recaptcha/api";
2. Change
const RECAPTCHA_VERIFY_SERVER = "api-verify.recaptcha.net";
to
const RECAPTCHA_VERIFY_SERVER = "www.google.com";
3. Change
self::RECAPTCHA_VERIFY_SERVER, "/verify",
to
self::RECAPTCHA_VERIFY_SERVER, "/recaptcha/api/verify",