I have a similar problem as many. One of my real friends (a live human being), does not have an Add Friend button on her page. I've never been Facebook friends with her and also never blocked her. Is it possible she blocked me? Is there a setting she added that caused this?
Yes, this is possible that she has blocked you and then unblocked you. Blocking/unblocking or denying a friend request can cause this problem. A person who receive request has always an option to deny request and report as a spam. Reporting spam make sure that sender will never be able to send request again to that person.
There are mainly three reason to not seeing "Add Friend" button:
- As Sathya has mentioned in his answer, setting "Who can contact me" as "Friends of friends" and you both have not any mutual friends.
- As I have mentioned above blocking/unblocking or denying request.
- A person has reached 5000 friends limit and now cannot add more.
Your friend still can send a friend request to you except in third case.
If these are not your case, please report a problem to Facebook.