I build a 5x5x5 led cube several years ago using a standalone 328p chip. The basic cube was driven using just 8 arduino output pins, 5 to control the 5 LED levels and 3 to control the two series connected 16 bit shift registers/constant current driver chips. So a 8x8x8 cube could be built using 11 output pins and use 8 series connected shift registers. The key is that the shift registers are wired in series so they all together only require 3 arduino output pins, not 3 output pins per shift register.
The hardware was based on a published circuit I found on the web. It shows using a PIC chip but the AVR chip does work just as well but with different software of course. Below is a link to the schematic drawing showing the hardware design. I highly recommend the use of the special shift registers that combine the data shifting along with constant current driver output pins, this saves using 25 (64 for a 8x8x8 cube) series current limiting resistors and makes adjustment of brightness of the leds very simple by just using two 'programming resistors' (one for each shift register).
http://picprojects.org.uk/projects/lc/Cube555Csch.pdf
Lefty