Why do airplanes have complex air conditioning systems?
It's because for the Air Cycle process, to chill ambient air without refrigerant, to work, you have to take really hot high pressure air and and remove most of the heat while keeping the pressure up, so that when you let the pressure drop at the end of the process it ends up way below ambient temperature.

So you take air off the engine compressor at some pressure like 80 psi and, say, 400F, pass it through a radiator (precooler) to cool it down, pass it though another compressor (the Air Cycle Machine's compressor) to boost the pressure, and therefore the temperature, again, run that through more air to air radiators or heat exchangers to cool that down, then, when you have the pressure still really high but have removed most of the heat, you let it expand and drop the pressure closer to ambient.  While doing that, you pass it by a turbine, which drives the compressor.  (An Air Cycle Machine is more or less a big turbocharger like device, pipes, radiators, and valves). 

You now end up with air that entered the engine at 70 deg F coming out the Air Cycle Machine at just above freezing when it's properly regulated.  That air is mixed with raw hot bleed downstream to create the final inflow to the cabin.  

An ACM can in theory put out extremely cold sub zero air if all the bleed flow available is run through it.  That is to be avoided because ice formation on the expansion turbine and inlet nozzles becomes a problem and this is one of the main factors in damaging them.  The other headache is dealing with all the condensate water, which causes turbine erosion problems.