Concept:
If you cascade identical amplifiers, then an increasing input signal will first saturate the last amp, then the amp before that, and so on. Now if you take the output of each amplifier and send this to an adjustable output summator, you can create all kinds of overload behaviour. You can see/hear this in the demo-movie:
https://youtu.be/f3LLiInagCY

Concept implementation (see: basic panel drawing):
To prevent getting a too complex user interface, I simplified things a bit.
Four amplifiers are used and each has a 3-position switch that allows the user to add-ignore-subtract the signal from it. In the circuit shown, you could also use 100k potentiometers instead of these switches if you want more adjustment possibilities.

Circuit implementation (see: basic circuit diagram):
To keep things simple and cheap I used one chip (<1Euro) that contains six amplifiers:  the hex CMOS inverter 4049UB (UB=unbuffered version).
These six unbuffered logic gates are actually high-gain inverting amplifiers that can very well be used for analog purposes. This chip is widely available and active in production, even in DIL-package. Adding negative feedback using resistors sets the gain. Cascading then four of these amplifiers gives a circuit (the pipeline) where a rising input level sequentially saturates: amp4, amp4+3, amp 4+3+2, amp 4+3+2+1.
The gain of amp1 (U1A) is made variable, being the input gain. Gain of amps 2,3,4 was chosen to be fixed 3V/V.
To allow both addition and subtraction of these four amp signals, each amp signal can be routed via a summing analog inverter (U1E) or can be directly send to the overall summing amp, the output variable-gain stage (U1F). All amps are build from the six gates in one 4049UB chip.

Circuit additions: odd/even harmonics control and modulator:
If saturation occurs symmetric, you create odd harmonics, so a potmeter labelled "symmetry" has been added that allows the user to also tune in asymmetry and thereby have more even harmonics.
To make the effect more lively, a low frequency oscillator (LFO) is added that can slowly sweep the symmetry setting. The LFO has a "depth" and a "rate" potmeter, the resulting modulation signal is made visible with a LED. A dual CMOS opamp chip makes the LFO, see: full circuit diagram.

The total circuit runs best on a supply voltage of 4 to 5V and draws only a few mA, so it is feasible to use a battery supply. The circuit has been build on a breadboard. 
At the moment (June 2024) I am working on a PCB design that I will post as an update when finished and tested..