What would happen if put a jet engine behind another jet engine?
There is an alternate way to answer your question, and it relates to thermodynamics.  

Think about the purpose of a jet engine.  It exists to take uncompressed air, compress it, mix it with fuel, burn it efficiently and then expand it (hopefully adiabatically) to generate a larger volume of gas at higher pressure than the intake.  The work done by the engine is described by the Brayton cycle (Figure 3.13 from http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node28.html), and this cycle shows us where we can and can't hope to achieve theoretically efficient operation.

If you look at the chart of the cycle in the above reference, you'll note that the compression and exhaustion portions of the cycle are where inefficiency (in the form of entropy) creep in and prevent the cycle from being perfect.  As such, even if there were sufficient oxygen left to burn, putting two jet engines back to back would be horribly inefficient (because you are repeating the inefficient portions of each twice).

The better solution is to work towards improving compression ratios, and recovering more work from expansion.  Much of this has been done over the last 50 years, and is why Brayton efficiencies are in the 0.6 to 0.7 range for modern jet engines (ref same website, Figure 3.19).

EDIT: Also, remember that jet engines are designed to work on air that is "cold" relative to their exhaust. They don't compress hot gases efficiently. You could certainly design a compressor to efficiently compress hot gases for combustion, but jet engine compressor sections are not designed for this, so the compression achieved by the second engine in your question would be MUCH lower than its design, which would further decrease it's contribution of additional thrust.