So, it's a fair question, but the 2nd law of thermodynamics quickly shows why this is not ideal.
So, the gas engine combusts fuel to release its energy. The energy added by the combustion raises the temperature of the combustion gases. Those high temperature combustion gases drive the turbine. As the combustion gases expand (in either a piston or a turbine), it decreases in pressure and temperature (these expansion processes are usually somewhat close to isentropic). In any case, the gases leaving the piston or tubine are a significantly lower pressures and temperatures than the initial gas after combustion. Looking at the Carnot equation, we are able to determine the maximum possible efficiencies of each gas.
Carnot efficiency = (1-TL/TH)*100%
So, let's assume after the combustion, the gas is around 2300 K, and leaves the piston/turbine around 500 K.
Carnot efficiency = (1-500K/2300K)*100% = 78.3%
Now let's assume we want to use the leftover gas (some of which is water vapor, as it is one of the reactants, which is what I gather you're asking about), then the temperature
of the leftover gases is at 500 K and ambient temperature is at 300 K, and the gases cannot undergo a process that drops them below ambient without having additional energy added to the process, which would decrease efficiency.
Carnot efficiency = (1-300K/500K)*100% = 40%
Now, bearing in mind that the Carnot efficiency is an irreversible cycle, in other words, no additional lost heat during the cycle (physically impossible as heat conducts through the engine and convects to the ambient air) and no friction (also impossible, as all mechanical processes are subject to friction), and bearing in mind that changing from 500 K to 300 K is usually a far less amount of energy than changing from 2300 K to 500 K, a change of 200 K as opposed to a change of 1800 K, the secondary cycle is rarely worth the harvesting. We call the heat rejected at 500 K "waste heat," as it is not nearly as useful as the gases immediately after the combustion. Conversion of waste heat to work is far less efficient (almost half that of the primary cycle), and the quantities of energy from the waste heat are also smaller.
So, in terms of cost, it is rarely justified to harvest this heat. Thermoelectric generators are sometimes applied to cases like these, but they are rarely worth the cost. Automobiles sometimes use a turbocharger that harvests exhaust fumes to drive a turbine that drives a compressor that increases the pressure to the air intake of the vehicle. Anyway, you can havest the energy, but it is usually not worth it for most applications, and you're usually better off designing your turbine or piston so that the gas leaving the system is as close to ambient pressure and temperature as possible upon exiting.
I hope that helped a bit.