What is the maximum temprature that could be achieved by compressing air?

There is no set temperature that is the default maximum threshold obtainable via pneumatic compression as @Tiger Guy and @Transistor said. You could, however, think of a compressed air tank much like a bomb calorimeter, in that the air stored inside it represents the system, and the outside room/atmosphere the surroundings. The system in this case generates very little measurable heat that is not absorbed immediately by the metal tank and the surroundings.

There is no "maximum," short of creating plasma. The temperature obtained will be a function of compression ratio and insulation. Other than that it is a simple T-P ratio (use kelvin or Rankine units).

Whether you can build a 400:1 compression ratio compressor is quite a different question.