Since taking an extreme consequence requires replacing an aspect, and since that aspect is permanent and requires milestones to change, what would you think of a character starting play with an aspect from a "healed" extreme consequence?
Something like MY BACK WAS TORN UP IN 'NAM. He was in the military during the Vietnam War and took shrapnel in the back, it paralyzed him for awhile. He has wizards constitution or one of the toughness abilities and it has started healing (I'm thinking of Harry's hand here.) By the time the game starts he is no longer paralyzed, but he still remembers it.
Or (keeping the Vietnam theme) Guest of the Hanoi Hilton, being a mental extreme consequence. Something has allowed him to start recovering, but it was still an extreme consequence at the time.
Something more recent could be Human Speed Bump. He has been hit by a car and it did some serious damage to a portion of his body. With age and/or a toughness ability it has started to heal, or he has accepted it, and its no longer an extreme consequence.
Politicians could probably be ripe for extreme social consequence. Events that killed their public opinion at the time. Clinton's "I did not have sex with that woman" springs to mind.
Maybe I'm thinking too hard, but it seems like another place to look for aspects during character creation.