The limiting factor in most jet engines is the heat at the front of the turbine, at the end of the flame cans. This is a problem when throttling the engine up, the fuel can be added quite quickly, before the engine itself is sped up and is supplying more air from the compressor. Heat builds up quickly and can cause the turbine blades to fail. This was a serious problem for the Germans, where engines averaged only 10 hours of operation before failing, often with chunks of metal flying out the back of the engine.
The solution to this problem was to bleed off some of the air from the compressor, run it down the shaft, and blow it through the middle of hollow turbine blades. This made the blades quite expensive to build. For a time jet engines often included the ability to inject water onto the engine to cool the exhaust in these cases. This was particularly notable because of the huge amounts of smoke that would pour out of the engine when it was turned on (typically for takeoff). A combination of better throttle controls and high-temperature alloys has removed the need for a water system.
The compressor uses up about 60 to 65% of all of the power generated by a jet engine. This explains why they aren't used in cars, you would be burning the fuel needed for a race while sitting still at a red light. Every bit of efficiency in running the compressor is needed, so one common design technique is to use more than one turbine to drive the compressors at various speeds. Most such designs use two stages, are are known as "two spool" engines.
Given that 60% of the engine's power is being used up for driving the compressor, one option for better efficiency is to do less compression - that is, make a smaller engine. This seems self-defeating, but it's not the case. If you instead use some of that energy not to compress the air, but simply push it, you can get thrust without compression.