K[n1_, n2_] := Union[{n1 + n2, n1 - n2, n1*n2, n1/n2}];
KK[list_, num_] := Union[Flatten[Table[K[list[[i]], num], {i, 
      1, Length[list]}]]];
KKK[list1_,
     list2_] := 
      Union[Flatten[Table[KK[list1, list2[[i]]], {i, 1, Length[list2]}]]];
d[a_, b_, c_, d_, f_] := {
    (*abcdfff*)
    f[{a}, f[{b}, f[{c}, {d}]]],
    (*abcfdff*)
    f[{a}, f[f[{b}, {c}], {d}]],
    (*abcffdf*)
    f[f[{a}, f[{b}, {c}]], {d}],
    (*abfcdff*)
    f[f[{a}, {b}], f[{c}, {d}]],
    (*abfcfdf*)
    f[f[f[{a}, {b}], {c}], {d}]
    }
d[l_, f_] := d[l[[1]], l[[2]], l[[3]], l[[4]], f];
dd[l_, f_] := dd[l[[1]], l[[2]], l[[3]], l[[4]], f];
dd[a_, b_, c_, d_, f_] :=
  (
    Print["abcdfff:", f[{a}, f[{b}, f[{c}, {d}]]]];
    Print["abcfdff:", f[{a}, f[f[{b}, {c}], {d}]]];
    Print["abcffdf:", f[f[{a}, f[{b}, {c}]], {d}]];
    Print["abfcdff:", f[f[{a}, {b}], f[{c}, {d}]]];
    Print["abfcfdf:", f[f[f[{a}, {b}], {c}], {d}]];
    )
p = Permutations[{3, 3, 8, 8}];
res = Table[Union[Flatten[d[p[[i]], KKK]]], {i, 1, Length[p]}];
Union[Flatten[res]]