The Rubik's Cube has so many well-known solutions that I thought it would be fun here to document a rediculous but (to me) interesting solution that operates under the constraint that one is only allowed to perform a basic commutator across adjacent faces rather than being able to rotate any face at any time. To be precise, the solution sequence must be of the form A,B,C,D,... where each of these is a commutator formed from any two faces of the cube. The only exception to this rule, given in step 1 below, will illustrate nicely the concept of parity that can be found in many twisty puzzles.
Begin by solving the edge cubie positions. These are tri-cycled by sequences of the form A,B,Ai,Bi, where A and B are each any one of R,Ri,L,Li,U,Ui,D,Di,F,Fi,B,Bi, provided the faces in question are adjacent to one another. If you find that the last two edge cubies must be swapped, you have parity. Solving this parity requires the only departure we need make from the constraint given above. After rotate any desired face a quater turn, it then becomes possible for this step to be completed. Of course, choosing the right face will minimize the amount of work needed to complete the step.
Now solve the edge cubie orientations. You can do this with X,Y,Xi,Yi where X=Ri,F,R,Fi and Y=|RUF|i. If you were careful enough in step 1, this sequence will need only be applied a few times.
Solve corner cubie positions. Notice that pairs of swaps are performed by sequences of the form 3[A,B,Ai,Bi] where A and B are as stated in step 1.
Solve corner cubie orientations. Use 3[3X,Y] to rotate 3 corners at a time on a single face. This is time-consuming, yes, but serves to illustrate that we can get the cube solved under the constraints outlined above.
I should at least mention a few other notable solution methods of the Rubik's cube.
Layer method comes to mind first. There are essentially three steps: first layer, second layer, third. The last layer has been studied theroughly by speed cubists and can be broken down into two sub-steps: OLL and PLL. The first two steps are are collectively referred to as F2L, and usually broken down, not by layer, but by the cross step, and then the remainder.
Next is key-hole method. Corners are solved first, then three of the four edge cubies on each of two opposite sides are solved leaving a notch or key-hole on both sides. The last two unsolved edges across both of those sides are then solved simultaneously. The last step is to position and orient the remaining edge cubies about the middle slice.
For any face A, letting A' denote the adjacent middle slice, we have R'=L'i, U'=D'i and F'=B'i. Super-flip is then given by 3[3[R',Ui],|R|,|U|]. This flips all the edge cubies.
The snake pattern begins with R,2F'i,Li,2U,R',2U,R'i,2|R|,R',2U,R'i,2U. Now just flip two edges on opposite sides of the cube to complete the pattern. You can easily get snake-in-snake on an NxNxN if you do the nested snakes in the right order.
Cube-in-a-cube is easy to get on an NxNxN as it easily generalizes from what we do to get it on the 3x3x3. Start with 4[R',U'i] to do a pair of center tri-cycles. Now after a setup move, tri-cycle the needed edge cubies on one corner, then the opposite corner, accounting for proper orientation. The sequence used is A,B,Ai,Bi where A=R,U,Ri,Ui and B=Li,Ui,L,U. Now twist corners on opposite sides of the cube. A setup move is useful for this.