Can a self-learning appliance be developed? I have heard of machine learning systems that can learn from trials of data what effect it should have, but I was wondering is it possible to make a learning appliance that builds on past data? Instead of only receiving new entries, once the database is big enough to begin going over it and doing pseudo consideration, drawing conclusions about the data and adding yet more entries to its database purely from looking at its past results? I suppose such a thing would probably eat up memory quickly unless it had a threshold of entries it was allowed to create. (In theory it would take a long time to get it to a state where it can become sentient in that way, through many trial sets before it begins thinking on its own.)

An application of such a thing would go beyond just determining what the best way to optimize code is, or how to beat a player in a game, but a machine that can actually build its own personality.
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.