Here's a SPICE model where each NPN stage is biased by its three resistors in such a way that none of the stages can saturate no matter how high the transistor gain ß gets.
I'm showing the SPICE model for a linear amplifier that I worked on. I needed a linear amplifier with very low level input signals at approximately 40kHz, the operating frequency of an ultrasonic transducer, and I wanted lots and lots and lots of gain at very low cost and with assured stability.
Each NPN stage is biased by its three resistors in such a way that none of the stages can saturate no matter how high the transistor gain ß gets to be and of course, none of them can ever be in cut-off. Ergo, each stage is guaranteed to be an active gain element. Also, since there is no signal feedback as would be the case with op-amps, these stages cannot oscillate. Their non-oscillatory stability is guaranteed.
The operational result is as follows:
The nice thing about these designs is that there is no intrinsic limit as to how many stages can be included. I used four stages and got more than 100dB of gain which was plenty, but there could be five stages, six stages… with virtually no limit other than perhaps input noise eventually driving the last stage to an undesirable output noise level.
There is some device dependency though. Going from 2N3904 transistors to 2N2222 in the SPICE simulation, we can see that dependency:
The completed ultrasonic detection system was hypersensitive and yes, false alarms were an issue. With a friend watching an oscilloscope, I crawled under a workbench (I was young and skinny then) and we both stayed stock-still. Then when I moved my fingers just slightly, the scope trace jumped.