Fully Functional 3x3x4
When a puzzle is described as fully functional, it means that it turns on all of the axes that you might expect. That is the case with the fully functional 3x3x4. Until quite recently, this puzzle was something that would be handcrafted by people that have a talent for modifying cubes. Tony Fisher and Katsuhiko Okamoto both had versions of this puzzle long before they were mass-produced.
As well as having more pieces to solve than a standard cube, the 3x3x4 also restricts the turns you can do on certain faces - you can only do quarter turns on the middle layer slices. As with a domino, that reduces the options you have for solving this puzzle.
There are various approaches to solving this puzzle, most involve holding it as shown in the photo, with the 3x3x3 face on top. You can treat it like 2 domino cubes, one inside another - solve the inner domino and then do the outer domino, trying not to mess up the inside as you go. Another approach is to solve the top and bottom layers first and then do the middle layer edges, then the centres. You can get parity errors on this puzzle - swapping the place of the centres tends to deal with it - or you can look up an algorithm.
These puzzles are fairly easy to find online. The one on the left of the photo is a DIY kit from Cube4You. It took a while to put together (you never get instructions with DIY kits) but turns beautifully. The one on the right of the photo came assembled from Puzl and also turns nicely. The fully functional 3x3x4 is an interesting puzzle to solve and a lovely addition to the collection.