dominicgs had some good ideas since software defined radio is very powerful. One extreme example of something you could do with a reconfigurable radio signal processor in your machine is install things that the FCC would not let be produced and sold. For example if someone reversed the military high accuracy GPS stuff, and produced a verilog spec anonymously so he was not sent to prison forever, no one would be able to use that to produce chips in consumer devices. But if we found verilog and wanted military GPS on our phones, all we do is install software.
Another example some friends have been excited about is emulation of older hardware, including video games, but much more. I will describe old games since they are closer to people's hearts. All the old cartridge based games had the actual cartridge contain half of the hardware needed to make a full computer. This immediately complicated emulation. But the real issue comes from quirks in the actual hardware. This is an issue because developers had no API except the exact hardware specifications that every console had. No one worried about checking their clock frequency was correct and dividing it to the right point so they got a certain framerate... everything was constant. Developers would find and exploit undocumented glitches in the hardware to make things faster, or sometimes to make them run at all. Hell, some games used the difference in clock frequency between two internal chips as a source for PCM sound to make little bleeps so they could save code. For this last example this means that unless you are emulating to the clock cycle each of these chips and sometimes their exact internal structures, some game will not work or crash randomly. The CPU power to do this is insane: a 3GHz core was the minimum suggested processor for running a highly accurate NES emulator in real time.
FPGAs let us implement the actual system, bugs and all, in hardware and not worry about CPU usage. Give the fpga a blob of shared memory for writing video and PCM data to and the CPU just passes blobs of memory between devices.
Programs like Postgres could implement highly optimized methods of certain algorithms like hashing for looking up in an index.
The kernel could detect it has FPGA space and offload some of its work to the chip.
I hope these have sounded interesting.