There is no way around this. If the friend request button is missing on a timeline of a person you wish to add as a friend, your only option is to contact that person some other way (like calling them on the phone). Since you would never break the rules by trying to add someone you do not know in real life, you surely have their phone number or have some other way of contacting them, right? Contact them some other way, and have them send YOU a request. That will still be possible, as YOU have never denied THEIR request to be Facebook friends.
Most people do not realize that this has been an option for quite a while. Up until recently, when someone denies a friend request, they are asked by Facebook if they know the person in real life or not. If the person answers 'no' then the friend request button is removed from view for that person to prevent that person from harassing them by sending another request. Facebook now makes this mandatory and will not let people send another request automatically.