What is the simplest sound generating circuit?

The 555 is a good way to start making tones in a speaker. I suggest you make a simple oscillator using one, before you attack projects that use several of them.

Also, we had a question, What is the simplest way to make an oscillating signal? That turned out to be an inverter gate with feedback.

Super-simple solution: a Schmitt's trigger inverter like this:

and you create a simple square wave tone. You can also filter and manipulate it to make different waveforms.

Consider that with oscillators at different frequencies you can create virtually every sound: that's what the Hammond organ does.

A slightly more complicated but extremely more versatile solution is to use a microcontroller with a quick DAC and generate tones with it. Then you can do many things, from using pots to set frequency and volume, but also create loops or more sophisticated things.

Sound is just vibrations, vibrations are caused by "blinking" the speaker coil. The only difference is instead of blinking a few times a second, to blink a few hundred times a second. Whatever you did to make the LED blink, change out some capacitors with smaller ones, and stick the speaker in place of the LED. It might make noise.