What about playing a character where you have the best of one attribute the worst of the other until all attributes are filled. That can be very entertaining. For example, 18 strength but 3 dexterity. 18 intelligence but 3 wisdom, etc.
So I've never gone to this extreme, but I have been GM when someone played a character with a very high dexterity and low (Around six) strength that was a… wait for it… fighter. Yup, and he played it well. He'd say things like, "Strength is only for the weak." He built everything around his dexterity. What a great memory.
While I have not had a character like this I had a friend who played a half orc with a 18 Strength and a 3 constitution. It was very interesting to say the least. Every time he got hit he had to save or fall down in agony.
For me the game will end up moving very slow or rather the main goal will be hard to accomplish because very low attributes will cause your character to do something really stupid or be the brunt of some social dislike that you always have to resolve things before you can even get on with the adventure. Imagine always breaking things, or insulting the barkeeper or being hideous to look at. All those things will draw a negative response.
This will make for a very funny game but as a Dungeon Master it will be hard to track all the mishaps that come from such low scores. There is plenty room for everything to go wrong and for all NPCs to react to it.
Yeah, having one character with low extreme scores isn't bad, and then only if a couple are extreme, not all of them. But to have multiple characters like this would be a horror as the Dungeon Master. I mean, you'd have to make sure the players role play their characters correctly and keeping up with that would be a bear.