Higher order cubes have been made, so why not a higher order Megaminx? Well, a very small number have been made by puzzle enhusiasts, but if you really want to do one you'll have to do it on computer. In keeping with the SI prefixes, the puzzles has been named the Gigaminx.
1. Where can I do a Gigaminx?
Twisty Polyhedra Applets has lots of great puzzles and a good interface. Unfortunately there are still a few major bugs but the program is being constantly updated so these should be worked out soon enough.
2. How should I go about solving a Gigaminx?
The Gigaminx can be approached with the reduction method in the same way as the 4x4x4 and 5x5x5 cubes. I'd recommend Chris Hardwick's solution to get started with the reduction method if you aren't aquainted with it already.
3. Are there parity issues on the Gigaminx?
Actually no. Although you might expect it have parity issues like the 5x5x5 it actually doesn't, because both inner and outer slice turns are even permutations. On a cube, inner slice turns are odd permutations, which is what causes parity errors. Visit Jaap's Puzzle Page and read the "Useful Mathematics" article for a better explanation of parity and permutation groups as applied to twisty puzzles.
4. How long does the Gigaminx take to solve?
The Gigaminx can be solved in less than ten minutes, but for your first time you should make sure you have a free hour or more. Alot depends on how well you work with a computer puzzle. My first attempt took an hour and nine minutes, but the puzzle was very tough to turn at times. People who are more experienced with computer simulations and doing large puzzles will probably only need 30 minutes or less.
5. In what order did you solve the faces and edge groups?
For the faces, I began with two opposite faces. Then I solved a ring of five faces around one of the first two faces. The last five faces are easier if they lie in one ring. With the edge groups, I solved the five edges around the bottom and put them in the bottom layer, then five more edges and put them in the next five lowermost spots. After that I solved the ten "equatorial" edges, then the five edges on the top, then the last five edges. The idea behind working "up" the puzzle is to gradually clear parts of the puzzle of unsolved pieces, so you won't have to look around as much to find the pieces you need and setup moves will be easier.
6. The Gigaminx is a long puzzle and it isn't very hard. Why solve it?
For the same reason people solve 20x20x20 cubes, of course. But seriously, solving the Gigaminx actually showed me some ways in which I could improve my 4x4x4 method, and would probably help me with the 5x5x5 even more if I were interested in solving the puzzle for speed. The Minx puzzles' similarities to cubes mean that improvement on one puzzle or the other often translates into improvements on both, and the Minxes seem to especially help one learn to find pieces more quickly.