What is the correct punctuation for a final "final exam" that has been taken? Should the sentence read: 'That my was my final, final exam!' or 'That was my final final exam!'
Omitting the comma is appropriate, assuming you are referring to the last of a sequence of final exams. In that case, it generally would be better to use an alternative word for the first final, as in my last final exam.
I'm assuming you're purposefully using the word final consecutively, even though it may cause the reader to do a bit of a double-take. If so, there are a couple of ways you could make it more readable, such as:
By putting the first "final" in italics, you manage to offset it from the word "final" in "final exam". (This would be especially effective if you wanted to emphasize some strong emotion, too, such as a sense of relief or accomplishment after completing your last final exam.)
Or, you might get away with:
By putting "Final Exam" in title case, it's obvious that the first "final" modifies "Final Exam". (The words "final exam" wouldn't normally be capitalized, but, in this instance, I think you can get away with doing so, as it could be considered the title of a standard element of an academic course – along with the Course Syllabus, and the Midterm Exam.)
I think either of these would be better than putting a comma between final and final. However, as others have said, you can also leave it as is – no comma, no italics, no title case – and it reads just fine. And that's my final "final answer."