Does this type of amplifier, IC or component exist?
What you are looking for is called Automatic Gain Control (AGC).  Many audio ADCs have AGC (I've personally used the PCM1870).

If you wish to use your MCU's ADC, I can think of a few options:


Use an external AGC, like the NE570, or build it using op amps (a cursory web search revealed that there are several reference circuits available).
Use the MCU to assess amplitude/clipping and find a way to control the ADC's voltage reference.  This option gives you the most flexibility.  You could use a DAC or a digital potentiometer.
Same as the above but instead of controlling the ADC's ref voltage, set the gain of a variable gain amplifier (VGA).
Use multiple fixed gain paths fed into the ADC's mux, like Olin suggested.  This will probably use more area and possibly be more expensive than a single variable amp, depending on how many fixed gains you find good enough.
Limit your AGC to digital amplification inside the MCU.  This probably yields the worst quality for low volumes, but it is the easiest and cheapest, and it may be good enough for whatever it is you're doing.  The viability of this depends on the effective resolution of your MCU's ADC, and how low a volume you are willing to tolerate.  


As a footnote let me warn you that if you roll your own gain logic/controller, adjustments must be made slowly and care must be taken as to not introduce audible artifacts or distortion.