Is there an audio amplifier chip or circuit that amplifies certain amplitudes in a signal less or more than other amplitudes?
This sounds like an audio compressor.

An Audio compressor is a device that passes a low-level signal unchanged but begins to reduce gain when the audio level rises above the threshold value.

The amount of gain reduction is expressed as a ratio: a 2:1 slope allows the output signal to increase by only by 1 dB for every 2 dB increase of input level.

When the slope approaches 10:1 or more, the compressor functions more like a limiter.

There are a lot of nuances with compressors and limiters.  Slope, how the compressor behaves at the threshold point (easy-over or not).

dbx corporation used to make some of the best analog compressors on the market.  The core technology has been spun off into a new company called "THAT Corporation".  THAT Corporation.  In my humble opinion, they make some of the best audio components available on the planet.

I've used analog compressors and limiters from a number of manufacturers: the dbx units range from good to awesomely-good and the Drawmer units that I've used were even better. Drawmer

There is now a lot of digital implementations out there.  Pretty much every digital console manufacturer has decent compressors and limiters all done in the digital domain.  

I'm an old-school audio person and haven't spent all that much time on various digital consoles.  But they all seem pretty good: Yamaha, DigiDesign, Digico, Calrec.