If you have redirections in your site document root directory, then you could have a problem regarding mishandled links in wp-reCAPTCHA PHP code.
I had a recent redirection problem with wp-reCAPTCHA. The options form pointed to $_SERVER['PHP_SELF'], who fetches the raw address of the script file and thus gets scrambled by actual site redirection, since form actions are delivered via http server (apache 2.0) and not directly by the OS. The actual page would never be reached and WP got lost in endless login forms.
I solved the issue hacking the plugin code (version 2.9.2, wp-recaptcha.php, line 696) and changing the options form action to $_SERVER['REDIRECT_SCRIPT_URI'].
Btw, latest versions of the plugin seem to have the same problem, and I also have alerted reCAPTCHA support team about the issue.