I am curious to know if one can amplify a signal or make nonlinear active distortion to an electric signal without semiconductor transistor or vacuum tube technologies?
Yes, there is for example the magnetic amplifier. It is neither a transformer nor a relay but abuses the saturation effect of transformer cores to control a high current using a smaller control current. I think this defines a active device.
They even built logic gates around this principle in the 60th.
Neon bulbs can be used for certain switching applications, based upon the fact that once they've ionized the holding voltage drops below the strike voltage. Additionally, neon tubes with multiple suitably-shaped leads can be used for things like "counter" functions. One nice feature of those is that the same tube can be used to both count pulses and show the count in human-visible form.
Yes. I read an web article several years ago that described a guitar wah-wah pedal that had a unique sound. Wah-wah pedals use a sweep filter activated by the pedal to boost the signal over a narrow band. The author reckoned that this particular model's characteristic sound was due to a custom inductor in the filter circuit. It appeared to give asymmetrical output and that this was due to the core being magnetised leading to core saturation at a lower level on, say, positive signals than on negative.
There is a high distortion amplifier that has a limited frequency response .It is over 100 years old .It was used for a repeater in long distance telephone lines.It was constructed from a solenoid with its armature directly attached to a carbon microphone .The carbon microphone itself can be considered as an amplifier.The carbon microphone came out of a design contract that the young Thomas Edison had to "Improve the telephone " Alexandra Graham Bell was getting old by then .I suppose you could attach a small heavy magnet speaker to a carbon microphone if you can find a carbon microphone .
I am surprised nobody has mentioned the important ongoing research towards creating a biological transistor. For granted, it cannot be (yet) called a technology, I suspect, until further improvements and cross-validation is done in the field.
The transcriptor, as described in the abstract of the original publication in Science (May 2013):
We developed a three-terminal device architecture, termed the transcriptor, that uses [...] We realized permanent amplifying AND, NAND, OR, XOR, NOR, and XNOR gates actuated across common control signal ranges and sequential logic supporting autonomous cell-cell communication of DNA encoding distinct logic-gate states.