How hot do internal parts of jet engine get?
This parameter is called Turbine Inlet Temperature (TIT)


  Turbine Inlet Temperature (TIT) is the temperature of the combustion
  chamber exhaust gases as they enter the turbine unit. The gas
  temperature is measured by a number of thermocouples mounted in the
  exhaust stream and is presented on a flight deck gauge in either
  degrees Fahrenheit or degrees Celcius.


Essentially it's the temperature going out of the combustion chamber, representing the highest temperature the turbine would face.

Here's a figure showing how we are doing recently. So we are at around 1800K now and may reach 2000K soon.


For the temperature of the actual part under cooling, here is the current status of materials: we are at around the 1350K mark and may approach 1400K soon.