I mean no rest breaks, no food stops, no sleep. Very short bathroom breaks would be ok.
I don't think UMCA keeps such records, but if you were to include supported rides (where the rider doesn't stop for food, but can receive supplies from a support vehicle) then you might want to check and see if RAAM keeps any records on who might have done the longest solo segment.
Madisons are ridden by two-man teams, occasionally three-man teams. The early rule was that a rider from each team had to be riding at all times. The photos of riders reading a paper while steering with a foot on the handlebars (at 4am) are worthwhile.
Two man teams were instituted in 1900 or so, as a result of a New York law preventing individual six day riders from racing 24 hours a day. That "serious mental impairment" was the impetus behind the law.
The early history of six day racing is quite fascinating, should anyone be interested. "Hearts of Lions" is a very good read on the subject.
I've got a copy of "Heart of Lions", a good book. The solo six day riders didn't have to be on the track all the time but, to win, they had to try. They didn't fit the OP's definition of no sleep or your description of "without putting a foot down". Even so, your response is probably better than mine. Wikipedia is pretty good on this topic. http://en.wikipedia.org/wiki/Six-day_racing
Another possibility is the beginning of RAAM. The frontrunners do the first couple of days without significant stops. The track riders probably went longer though (less things to run into on the velodrome).
Taking the OP a little further, I believe there is a record for riding without eating or drinking. I think it was Pete Penseyres with something like 267 miles, but I can't find a source. Anyone?
Was Danny Chew. 170 miles without eating or drinking. from Pittsburgh to State College PA.
Cited in The Complete Book of Long Distance Cycling. (and thanks to Google Book Search I was able to find it... I'm on the road and my copy is at home - but I knew I read it somewhere.)