Connecting a LED between two 74HC595 without current limiting resistor
My opinion? Bad idea. Whilst the internal mosfets of the logic device have a finite on resistance, relying on this is not wise as they will be thermally limited - the more current you flow through them, the hotter they get. The HC595 datasheet will outline these limitations. Sure, it will 'work', but it is still bad practice.
There's WS2812 (neopixel) leds and their variants that have an internal controller and driver. You can run a number of them off one port pin.