Can a self-learning appliance be developed?
You may be thinking of some combination of online and offline learning.

Online learning only use new examples to base its learning on, and modifies the model at each step.

Offline learning looks at all available data, then fixes its model and uses that model on new data.

There is some use in creating new examples from existing ones, but this is typically done by hand, as to emulate possible variations expected to appear in actual data (and typically only if you don't have enough data). There isn't much use in automating this, as any conclusions it draws from the data will already be reflected in the model - if we add entries based on these conclusions, and in turn use these to draw new conclusions, either the conclusions will remain exactly the same (i.e. there's no point in adding them), or we'll start to lean towards conclusions not corresponding to the original data, which is not a good idea.

The only way AI can build its own personality, is if you program the potential to do so into it. There's no way AI can suddenly become sentient after playing 1000000000000 games of chess if it's only been taught to play chess and only knows about chess.