74HC595 has 70mA Absolute max limit, so you were close PaulRB.
Agreed, TPIC6B595 is an awesome chip for combined shift register/high voltage (50V)/ high current sink (150mA continuous per IO). For things like sinking current from individual LEDs or sinking current from 12V LED strips, it is awesome.
I daisychained 20 of them to drive this 5 x 20 LED matrix: