There are Kontakt libraries that generate rhythms, sequences, and/or whole backing tracks with just a few clicks. They won't generate an entire piece, but they can provide a well-developed foundation on which to build further ideas.
Here are some developers that specialize in these types of libraries:
- Sample Logic: IMHO they have the best randomization engines and offer high quality sounds. Check out libraries like Arpology, Expeditions, Morphestra, Symphonic AI, etc.
- Audiofier: Practically the entire line is about one-finger music creation with plenty of randomization options. Check out the Veevum series, Xtyles, Tetrality, etc.
- Rigid Audio: Their Loop-Based libraries are basically a one-finger band that allow you to randomly mix and match among four musical parts to create backing arrangements. Libraries like Synferno, Cinematrix, Acoustic Isolation, etc. come with a surprisingly large amount of content. Just don't pay full price for these - they often go on sale for below $10 each.
There is also the Toontrack trifecta of EZdrummer, EZkeys, and EZbass. Although it requires a bit of concentrated effort, these tools can be used in conjunction with each other to generate complete fully-realized backing tracks. It's not a one-click solution, but the results are far more musical than most one-click gadgets. To broaden the range of what's possible, I'd recommend investing in EZkeys MIDI packs, since these provide the harmonic structure and musical stylings for the final arrangement.