Transfer Curve of Amplifier
Your cuve kinda looks like ań inverted tanh function.

That's the response a overdriven differential amplifier built from ordinary transistors would generate. 



Explanation from input to output:

The voltage divider built from R5 and R6 brings your input signal down into a range which overdrives the differential amplifier built from Q1 and Q2. As you can see we bring down the signal a lot. That's because the Q1/Q2 differential amp has a lot of gain to begin with.

Since we have only one signal we apply it to the base of Q1. The other input of the differential amplifier, the base of Q2 is grounded via R2.

We now see the following signals at the collectors of Q1 and Q2:


That's already close to what we want. We could pick the blue signal, amplify and level shift it. That would require quite a bit of calibration and tweaking to get working though.

It's easier to take the difference of the two signals. That's what the Opamp U1 and the surrounding resistors do. It's a very simple differential amplifier with a tiny bit of gain.

The resulting difference output looks like this:



That's pretty close to the waveform of your drawing. Note that I've used an ideal opamp model for simulation so the output waveform slightly goes above and below your supply voltage. That'll go away if you use a real part.

Some guidelines for tweaking:

Increase/Decrease R6 to control the amount of distortion you want. The higher you go, the more linear your signal will get.

The voltage dividers R9/R10 and R8/R7 set the gain. Their ratios should be identical.

If you want to built this circuit in practice:


all resistors should be 1% tolerance or better.
U1 should be an opamp with rail-to-rail output capability
Q1 and Q2 should ideally be matched transistors. It will work without if you can live with a bit of asymetry though. The actual NPN transistor does not matter much. Any jellybean part will do.