In my undergraduate experiences (in the US), I usually was able to, at the very least, take a look at my exam after it was graded. However, the exact practices seem to have varied by department (or even by instructor):
In two cases from the statistics department, I could take a look at the exam while in the professor's office, but I had to return it immediately afterwards.
In some cases (a physics and some economics classes), I was able to obtain the exam papers from a department secretary. I was also permitted to hold onto them.
In a third scenario (also economics), I was told to contact the TA's if I wanted my final exam papers. So not only were the TA's responsible for grading, but they also kept the portions of the exams they graded. However, I have no idea if in this case the papers eventually go back to the professors / department, should the TA's graduate.
In mathematics sequences taught by the same instructor, we simply received the final exam papers in class in the succeeding term. I am not sure if the papers for the final term of the course were retrievable: I would assume that they were but never attempted to access them.
In all of these cases, I was able to access my papers within two or three months of the exam. However, I also remember one instance where I had wanted to obtain the comments on a final paper for a history class. The graduate student who taught the class was unresponsive to my emails, but in our intermittent exchanges, did indicate that it would be possible for me to get comments for the final paper (which had been submitted as a hard copy) back even two terms after that class had finished. (This meant that he must have held onto the work in question. However, my institution also goes by the quarter system, so "two terms" is really only equivalent to about four or five months.)