Positive Mex

Chef recently read about the mex function and came up with a function similar to that. For a (possibly empty) sequence of positive integers $S$, Chef defines $f(S)$ as the smallest positive integer that does not appear in $S$. For example, $f([])=1$, $f([3,4,3,5])=1$, $f([2,5,1,1,2,3])=4$. Chef's friend heard about Chef's idea and found it really cool. Therefore, he sent a sequence $a_1, a_2, \