I can kinda sorta solve a 4x4x4 cube, known at the Fry's Marketplace as "Rubik's Revenge".
The most common way to do it is to solve the centers and then pair up the edges, and then solve like a 3x3x3, but there's a good chance you'll end up with parity problems that take a long time to solve, and I haven't memorized the ways to get out of those yet.
There seem to be alternate ways of solving it slightly differently, and some of them apply to the 5x5x5 cube as well. If I can figure that one out, then... the Gigaminx, perhaps?
The most common way to do it is to solve the centers and then pair up the edges, and then solve like a 3x3x3, but there's a good chance you'll end up with parity problems that take a long time to solve, and I haven't memorized the ways to get out of those yet.
There seem to be alternate ways of solving it slightly differently, and some of them apply to the 5x5x5 cube as well. If I can figure that one out, then... the Gigaminx, perhaps?