Does chaining Carnot heat engines make them more efficient? Since the efficiency of a Carnot cycle depends on the difference between the hot and cold sides of the engine, could you put multiple heat engines in a series to maximize overall efficiency? A $3000 \, \text{K} \to 2000 \, \text{K}$ engine connected to a $2000\, \text{K} \to 1000\, \text{K}$ engine connected to a $1000 \, \text{K} \to 500\, \text{K}$ engine, and so on?
 A: This is common practice in heat engines. For example, in large reciprocating steam engines, you'll have three pistons operating in series: a small, high-pressure piston, a medium-size mid-pressure piston, and finally a large, low-pressure piston. the exhaust from the first one is expanded again in the second one, and so on, with the inlet pressure and temperature falling in each of the stages. 
This is also done in large steam turbines, where each turbine wheel on the shaft is larger in diameter than the previous wheel and passes its exhaust on for more expansion in the next downstream stage. 
A: Yes, you certainly could do that. Assuming that each sub-engine was operating at the Carnot efficiency then the total efficiency would be equal to a single Carnot-efficiency engine operating between the extremes. Heat engines become less efficient the closer their input and output temperatures. 
So in your example, the chain of engines would be more efficient than a single 3000 to 2000 engine alone, but the same efficiency as a single 3000 to 500 engine
A: If you take out all the heat you put into the intermediate reservoirs, so that heat only flows on net from the hottest to the coldest, then it doesn’t make any difference. That is, the effect of the multiple engines “cancels out”, and you end up with the same efficiency as a Carnot engine run between the hottest and coldest reservoirs alone.
The easiest way to see this (without doing the calculation) is to note that the Carnot efficiency is the unique efficiency for all reversible engines. Since your setup is reversible, being made up of reversible Carnot engines, it has this same efficiency.
Of course in the real world, engines are not reversible, so the procedure you describe might improve efficiency in practice. To get a solid answer in that case, you’d have to be much more specific about the setup, and talk to engineers, not physicists.
A: This does not work in such a way.
Even if you increase the number of heat engines, all connected in series, the efficiency of the all the heat engines combined will still be 1-T(h)/T(c). Where T(h) is the temperature of the hottest reservoir and Tc is the temperature of the coldest reservoir.
Just google the derivation of the thermodynamic temperature scale and it'll give you more insight to what i said in the above paragraph.
In real life, you are probably gonna decrease the efficiency in such a way because the greater the number of engines will lead to more irreversibilities.
A: There is this thing about theoretical limits.  Whenever someone tells you that he has a completely novel approach that will, given enough grant money, yield better results than the theoretical optimum, hold onto your purse and run.
Combining multiple ideal machines will not yield better results than a single one.  While real machines may have an operating point they are optimized for, even at their operating point they will not beat the theoretical limit, nor be part of a construction that does.
Well-sealed Carnot engines (with regard to temperature and pressure) are as good as it gets.  They are also awfully slow.  Sequencing several machines may convert heat difference to mechanical energy faster, but when the heat is used up, you'll not have gotten out more mechanical work.
Cascading may well help with scaling down the size of the "well-sealed" problem.
