Yeah, they've become pretty aggressive when it comes to getting people to upgrade. Quite a few of us are kind of upset about the amount of spam that comes up in the free version. But, I'm in QA, and nobody listens to us.

Depending on what you're doing, there are a few different reasons it'll try to connect to the internet.

Ripping - Trying to connect to CDDB
Radio - It's internet radio, so obviously
Guide or Now Playing window open - Retrieve info from AMG
Listening to MP3s - Uploading of anonymous play logs

If you really don't want it to connect to the internet, you should be able to disable all this.

Go into the Now Playing window and click the "Online" button to take yourself offline. That should stop any song info retrieving.

You are given the option to opt-in or opt-out of playlog uploads. They're used for recommendations. If you don't want this, go to Options/Settings... and at the bottom of the General tab, there's a checkbox for uploading user pref info. You might also want to disable the periodic checking for updates (right above it).

Also in settings is the CDDB tab, which allows you to disable CDDB lookup.

I know that's a lot of stuff to do, but that should hopefully stop any internet connectivity.