Update:
I found what was keeping me from logging in on 3.8 and it a “coming soon” plugin on my site since it’s under development. Once I turned that off, I was able to connect on the mailchimp plugin.
Just tried on another machine – same results.
I click the connect button, a small window appears for the username and password, enter my info and press Log In, button says “Please Wait” for a few seconds, small window closes, main page appears to be reloading, and ends with the same connect button.
The problem seems to be something with WordPress 3.8. I just tried the plugin on a WordPress 3.7.1 install and while the login process appeared to get hung-up, I was able to go back into the mailchimp page through the dashboard and I was now logged in.
Same issue. This is my first time installing the plugin and I was not able to connect via Firefox, Chrome or Safari. Any updates on this?