Vertically stacked photodetectors based on printable organic semiconductors have considerable potential to deliver high-performance, low-cost, filterless color sensors, which are highly sought after for a host of emerging applications. To realize the full potential of this technology, however, it is critical to minimize the impact of the optical losses occurring at the electrodes, which reduce the photon flux reaching deep into the device stack. To this end, here we propose the use of organic photodiodes with photoconductive gain within a vertically stacked device architecture. We first realize polymer-based green- and red-sensitive photodiodes with external quantum efficiencies (EQEs) greater than 100 % and optimize their processing and spectral properties for color sensing. We then analyze their integration in a vertical device stack atop a blue-selective photodetector to achieve red-green-blue (RGB) color sensing. Finally, we examine the ultimate limit of this approach for multispectral/hyperspectral light sensing. The resulting large and balanced photoconversion efficiencies for all RGB color bands and beyond demonstrate the potential of vertically stacked organic photodetectors with gain for high-performance, low-cost color/multispectral/hyperspectral sensors and imagers.