The reason you need to dump the heat is because engines, by their definition, operate in a cycle. They return to a previous configuration before continuing on. So your solution of just using an unlimited number of one-shot devices is theoretically possible. It just wouldn't be called an engine. It also wouldn't be practical. One might, however, consider the big bang itself to be the ultimate one-shot device!
Engines also want this cycle to go in one direction, so we have to engineer them to do so. In theory, one could have a device which simply goes in either direction without dumping heat. We've built such devices -- little turbines that operate at the molecular scale where random molecular motion causes things to bump one way or another. However, we can't make them do work (we tried). To get work out of them, we need to know which way around the cycle they will go.
To make that happen, we target two equilibrium, rather than one. One equilibrium is at maximum entropy, such as at the fullest expansion of a piston. Once we get there, we need to reset the machine, targeting a second equilibrium (such as with a piston at its most compressed). As we do this, we have to dump the heat because this second equilibrium is not the highest entropy state with all that heat in the system. We have to get rid of the heat before this second equilibrium is achieved.
Now you are allowed to use the heat to drive another engine. This is called a multi-stage engine and they are used in many power plants. They can be more efficient than a single stage engine. However, the laws of thermodynamics provide a hard limit on how efficient they can be, no matter how many stages you use. The resulting limit on efficiency is defined by Carnot's Theorem, and depends on the temperatures of the hot source and the cold sink. (Note: only heat engines have this limit. Other devices, such as fuel cells, do not operate as a heat engine, so they can achieve higher efficiency)
The ultimate example of this is a Matiroshka Brain. This is a fantastic megastructure wrapped around a star to get as much usable work out of the fusion engine as possible. It is a massive heat engine which has a tremendously large number of stages (thousands to millions), where the waste heat from each stage is the hot source for the next stage. The result could theoretically get close to the ultimate ideal heat engine.
For a Matrioshka Brain around our star (the sun), we can calculate its efficiency. The sun is roughly 5800K on its surface, so that's our hot temperature. Our low temperature is the background radiation of empty space, which is a mighty frigid 2.725. Plugging this into Carnot's equation, $\eta_{max}=1-\frac{T_C}{T_H}$, we get a maximum efficiency of 99.95%. These brains can be amazingly efficient, but they can never avoid the slow march of entropy!
