Can you sue a company for sending unwanted e-mails that you've unsubscribed for several times over?

In short, yes you can. There are statutes to protect people from unwanted commercial e-mails. I If you don't have a local attorney who handles these matters, you can contact me. I have a practice in the Atlanta metro area.

Yes, there are statutes which make this illegal and give a person who continues to receive such emails a cause of action against the sender, although there are some safe harbors built into the statute (e.g. you can't sue when you have a previous existing business relationship with the sender). I suggest that you print out copies and/or create a record of all emails you receive from the sender in question, and also make a record of your efforts to unsubscribe. If you are receiving the emails on a system which automatically purges the information after a period of time, take steps to avoid losing information that you may need to prove your case.