Is it possible for any circuit to independently make its own trigger, sweep or oscillation without the aid of an AC sine wave?
Is it possible to create an electrical circuit that only has one initiating signal (the press of a button or the presence of DC voltage, say) that can cause itself to switch based on that one signal?


Yes. 

Do a Google image search for "oscillator circuit" to get all the examples you have time for. For more specific results, try "Pierce oscillator", "Hartley oscillator", "relaxation oscillator", "astable multivibrator", ...

In many of these circuits, there is a theoretical fixed operating point without oscillation, but it is unstable, so that even the inherent noise of current flowing through a resistor will eventually start them oscillating. Others are a bit harder to get started, but generally the act of turning their power source from off to on will start them.