TI-Basic (TI-84), ¹¹ 10 bytes

Repeat checkTmr(startTmr Disp 1 End 

the program can only end when the check checkTmr(startTmr happens between two seconds (checkTmr(startTmr) = 1 instead of 0). This way, any number of ones is possible (the shortest I got was 3 ones, the longest printed during approx 10 seconds)

Repeat always executes the loop the first time, so the empty string is not included (repeat until or do while not)

For calculators without time (TI-83), replace checkTmr(startTmr with rand>.9 but it's not true randomness (same output on each reset of the calculator)