Creating one tone or even three or four tones on a breadboard might not be difficult. Here's one circuit for creating one frequency:
Don't know though, how you would create two consecutive tones. As I see it, as soon as you want just a few consecutive tones with distinctive pitches and lengths (to form a melody), you really should use microcontrollers. If you wonder what lies inside this chip:
... which plays only one melody ("It's a Small World"), I bet it is a microcontroller construction with its own integrated crystal for keeping the time, its own ROM holding the melody, its own CPU like part reading and executing a simple program inside the ROM. Creating a chip playing another tune means only changing the part of the ROM, where the melody is stored, not inventing from scratch another IC chip with three legs.
If you really want to create whole melodies without the microcontroller, just with resistors, transistors and capacitors (and the piezo buzzer), I guess there are no breadboards large enough. Each soundable tone would have its own set of components for creating the right pitch and length. And when ending the tone, the circuit would trigger the next circuit for the next tone.