But you don't have to limit yourself to a sequence of all the same number. All sorts of patterns might be deemed just as remarkable: {1,2,3,4,5,6,7}, {7,6,5,4,3,2,1}, (36,35,34,33,32,31,30}, {2,4,6,8,10,12,14}, {1,2,1,2,1,2,1,2} .....
As I said, it's subjective, but I would define all of the above and many more as sequences that would 'cause as much of a stir' as {29,29,29,29,29,29,29}.
It's still going to be a very small set, compared to all the possibilities of similar length, but suppose there are 1,000 such sequences, then we might be looking at an event you should expect to encounter once every 10 years, as opposed to once every 10,000!