As the title asks, I wonder if the size() method in the LinkedList class takes amortized O(1) time or O(n) time.
It's O(1). You can google for the source code and you will come to such:
From http://www.docjar.com/html/api/java/util/LinkedList.java.html
All of the Collection classes I have looked at store a the size as a variable and don't iterate through everything to get it.