Series[f[x],{x,0,1}] // FullForm

(* SeriesData[x, 0, List[f[0], Derivative[1][f][0]], 0, 2, 1] *)

Series[f[x],{x,0,1}] // FullForm

(* SeriesData[x, 0, List[fRef[0], Derivative[1][fReg][0]], Derivative[2][fReg][0]], -1, 2, 1] *)

Series[f[x_], y__] ^:= Series[1/x, y] Series[fReg[x], y];
(* SeriesData[x, 0, List[f2[0], Derivative[1][f2][0]], -1, 1, 1] *)

Series[x f[x] + Cos[x], {x, 0, 2}]

(* SeriesData[x, 0, List[1, f[0], Plus[Rational[-1, 2], Derivative[1][f][0]]], 0, 3, 1] *)