What are examples of penalties for "self-plagiarism"? I graduated from university in June of 2015. I submitted an essay to a philosophy professor in my second year that was substantially the same as an essay I submitted to a TA the year before for one of the the same professor's first year classes. 

I wanted to use my research and essay content for the second year course because I was enthusiastic about that topic and wanted to use it again. I asked the professor if I could use textbooks and information/material studied from his first year course (which also wasn't a prerequisite for the second year course) and he said that was fine. I wasn't really aware of this "self-plagiarism" rule back then and I figured he was saying it would be fine to use the content of my old work from the previous year as well. I do remember changing some parts of the essay but the two had substantially the same gist and body. 

I just realized recently that I need written permission from the professor in those situations. 

Theoretically, if they could prove now that I committed some sort of academic misconduct for that back in my second year, (2013-2014, I've had my four year honours degree since June 2015 and am now working in my field) what would the penalties be? Would my degree be revoked and would I be required to take another course? Would they just adjust my course or assignment mark? 

I know it depends on the circumstances so I broke it down and let me know if there are any more details you need to know.
It depends on many factors. Examples of those factors: Was it a first offense, was it an undergraduate or graduate student, was it faculty member, was it for an assignment or a publication? The list of penalties can range from using the incident as an "object-lesson", a failing grade for the paper or course, a formal sanction, expulsion, loss of funding and/or termination. Each university usually has an Academic Integrity Office, so you could always consult yours to see what the penalties are based on the seriousness of the charge.