8x8x8 multiplexed LED cube with an Arduino Mega 2560

Yes, see http://www.aosmd.com/pdfs/datasheet/AO3401.pdf. They are rated at 4A continuous (although 3.7A would be a more realistic limit when driven from 5V, since that is the current at which Rds(on) is measured for Vgs=4.5V) and 1.4W power dissipation (although I wouldn't want to dissipate more than a few hundred mW in one).