Every positive integer appears at least once; if we consider an arbitrary integer $ k $, it can be written in the form $ \binom{k}{1} $. Almost all integers can be written twice, because $ \binom{k}{1} = \binom{k}{k-1} = k $, but when $ k-1 = 1 $, the two occurrences overlap into one. This takes place for $ k = 2 $, and $ 2 $ only appears once in Pascal's Triangle, because succeeding rows of Pascal's Triangle only contain $ 1 $'s and numbers greater than $ 2 $. Therefore, the minimum value of $ f(n) $ is $ \boxed{1} $.