I wonder if anything is known about the distribution of the sum that exceeds one? Starting with Art's code above, it seems a beta distribution provides a good fit.

data simsum;
simsize=1E5;
do i = 1 to simsize;
sum=0;
do j = 1 by 1 until(sum>1);
sum+ranuni(0);
end;
output;
end;
run;

proc univariate;
histogram sum / beta(theta=1 scale=1);
run;