My primary email address is going to be invalid soon. So I want to change my email address for my Google account. However, it seems that I can't change my email address and I can't even add a new email address. This is what it looks like:
NOTE: On a recent comment to another answer, the OP mentioned that their domain is gulli.com, a German domain. According to Wikipedia, on May 2018 the domain was sold.
The email addresses of Google Apps accounts, including those of Google Apps for ISP, can't be changed by the user, only by the organization administrators. They should have access to the administration panel for your organization by signing into http://admin.google.com.
It's worth to say that when the original revision of this answer was posted it was not possible to change the primary domain of an organization , so if the domain will be invalid soon, then the organization information (the information of all the organization accounts) should be migrated. For details about how to migrate Google Apps information go to Migrate data away from Google Apps - Google Apps Help.
Nowadays G Suite allows to change the main domain. For details see Change your primary domain
On my previous answer the OP posted a comment saying
It's my private Google account. I don't think we're talking about the same thing. I don't have a Google Apps account, just a regular Google account (GMail, PlayStore, ...)
I understand that they means that the primary email address has a gmail.com or googlemail.com domain. In this case the only way to change the primary email address is to remove Gmail from the account then adding it again is you still want to use Gmail, otherwise you could have your Google account with a third party email address.