It's even better than I thought...I can get 2 16-bit shift registers out of only three Arduino pins. So, I should be able to get 8 16-bit shift registers out of 12 pins.
That's a heck of a lot of pins, and they can be used for more than just lighting LEDs, they can be used as general purpose pins, just like the native arduino pins.
Thanks a lot for the help and suggestion, guys! 