Try
list={0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,
17711,28657,46368,75025,121393,196418,317811,514229,832040,1346269,2178309};
crush[n_]:=Module[{m=n},If[m>=100,While[m>=10,m=Total[IntegerDigits[m]]]];m]
Map[crush,list]
which returns
{0,1,1,2,3,5,8,13,21,34,55,89,9,8,8,7,6,4,1,5,6,2,8,1,9,1,1,2,3,5,8,4,3}
Test this carefully and study the documentation for each of those functions used.