uses
  FileCtrl;

procedure RemoveAll(path: string);
var
  sr: TSearchRec;
begin
  if FindFirst(path + '\*.*', faAnyFile, sr) = 0 then
  begin
    repeat
      if sr.Attr and faDirectory = 0 then
      begin
        DeleteFile(path + '\' + sr.name);
      end
      else
      begin
        if pos('.', sr.name) <= 0 then
          RemoveAll(path + '\' + sr.name);
      end;
    until
      FindNext(sr) <> 0;
  end;
  FindClose(sr);
  RemoveDirectory(PChar(path));
end;

procedure TForm1.Button1Click(Sender: TObject);
var
  dir: string;
begin
  if SelectDirectory('Удаление каталога', '', dir) then
    RemoveAll(dir);
end;