Автор: http://www.swissdelphicenter.ch

function GetDays(ADate: TDate): Extended;
 var
   FirstOfYear: TDateTime;
 begin
   FirstOfYear := EncodeDate(StrToInt(FormatDateTime('yyyy', now)) - 1, 12, 31);
   Result      := ADate - FirstOfYear;
 end;

 procedure TForm1.Button1Click(Sender: TObject);
 begin
   label1.Caption := 'Today is the ' + FloatToStr(GetDays(Date)) + '. day of the year';
 end;