My 3x3x3 and 5x5x5 LED cube

Nice job with the build and assembly Wondertiger.
What current are you seeing from the 74HC595 outputs?
Set an LED high and measure the voltage across a 220 resistor.
I'd expect something like 2.8V/220 = 12.7ma if your LEDs have a Vf of 2.2V and you're actually gettiing 5V out of the 74CH595.