Confusion about LiPo charging/capacity monitoring with micro-controller
... will it:
  


Rechargeable (but really, all) batteries have a limit as to how quickly their chemical reactions happen. Any energy that isn't used in the reaction turns into heat, so putting in more current than possible could result in permanent damage.


  
  ... without using any special components?
  


You need to measure the voltage difference across a shunt resistor. Since this voltage is very small, you will also need to amplify the difference so that it can be measured with any precision.

The ATmega328 can do neither differential measurement nor does it have any gain, therefore you will either need to use something like the INA250 in order to perform coulomb counting, or you will need to switch to an AVR that can do both (and find a suitable shunt resistor).


  
  [charging]
  


You can use voltage to detect the end state for charge, but you need to switch the charging circuit off so that you can measure it unimpeded. The 2V start point is arbitrary; it is simply used to show how the charging curve may look.

The charging circuit must control the voltage and current; the battery will always take what it can and turn what it can't into heat. You may also want to consider adding temperature sensing so that you don't try to charge the battery when it is too hot or too cold (cold can slow chemical reactions, causing heat damage to the battery).