While you may not be able to use Google Analytics for Wordpress.com sites, without upgrading, there are ways you can track what is going on with your sites.
One method is using the tracking features of Pretty Links, if you are using that plugin. Keep in mind some of the features are only available to Pro users.
Another method you should definitely be taking advantage of is Google Webmaster tools. In many ways this can be better for tracking long tail keywords than Google Analytics. Follow the steps below to get set up:
1. Go to https://www.google.com/webmasters/tools/home?hl=en. Register for a Google account if you don't already have one.
2. Register Domain & Click Add (input your wordpress.com domain name here)
3. On the next page select alternate method tab.
4. Select HTML tag
5. Copy Meta tag (leave this page open)
6. Go to your site /wp-admin. (example: https://greenliving2015.wordpress.com/wp-admin)
7. On the left navigation select Tools -> Available Tools - > scroll down to Google Webmaster Tools
8. Paste Meta Code in Google webmaster tools and save changes.
9. Go back to the Google webmaster tools page you left open from step 5. and select verify.
10. You should receive a message congratulations you have successfully verified ownership of “your domain.”
11. Select continue to go to the webmaster tools console and see available info and options for your site.