CuInSe2 films for photovoltaic applications grown under Cu-excess have been rarely investigated up to now. In general CuInSe2 solar cells use an overall Cu-poor absorber. In this work we argue that it is valuable to investigate Cu-rich solar cells, since all the basic material properties are better in Cu-rich absorbers. With less defects in the bulk and better transport properties it is somehow intriguing why devices with Cu-rich absorber perform less. We demonstrate that this can be attributed to the too high doping of these films. Such a high native doping leads to tunneling enhanced recombination and interface recombination, strongly affecting the devices performances. We demonstrate different attempts to overcome the problem of doping: at first a Cu-poor surface was grown on the Cu-rich absorbers which enables to decrease the doping in the space charge region, then to directly decrease the doping in the bulk, the influence of sodium content was investigated. Finally, here we show that different selenium activity during the absorber growth enables to decrease the doping of these films and to open thus a way to fully exploit the favorable properties of the Curich CuInSe2 films.