Fully Functional 3x3x6
This puzzle is one step up from the fully functional 3x3x4. If you have seen one of those, you will know most of what you need to about this puzzle. The puzzle in the picture is made by Witeden and was bought from Meffert's store.
In the same way that a fully functional 3x3x4 is like having 2 dominoes, one inside the other, this puzzle extends the idea by adding a further domino. Solving this puzzle is like solving 3 domino puzzles. You get the same parity issues as you do with the 3x3x4 (some of the centres can be swapped with centres of the same colour). The algorithms you would use to deal with this on the 3x3x4 can be used with this puzzle if you turn 2 inner layers each time you see a u or d move.
The puzzle in the picture is not quite as smooth as the Cube4You 3x3x4 but is still robust enough and easy enough to use to make a nice puzzle to solve.