Processing video (especially if you have weird requirements like 14 bit greyscale input): https://gregdavill.com/blog/2018/9/9/boson-camera-project-pa...
Experimenting with non-mainstream CPUs: https://www.thanassis.space/myowncpu.html
Usually if you need extremely fast i/o in the nanosecond order of magnitude then FPGA designs are a common design choice. An example of this are GPON network switches.
As mentioned in another comment video processing or any signal processing where the algorithms benefit from high parallelization is also an application where an FPGA would be a good fit.
The caveat though is that often FPGA/ASIC development is expensive and slow so a recent trend is to have a System-On-Chip with an FPGA area and multicore Microcontrollers. The idea with this is a hybrid design so that you can have an RTOS dealing with functionality where speed is not as critical. And have a custom design on an FPGA that is responsible for whatever bespoke application you need and have some memory interface between the two.
Upvote for this. When I used to think about PROM, I think it as a medium of data storage, or sometimes think it's a lookup table. But it's actually simplest form of programmable logic device - a device that can transform x-bit of arbitrary input signals to y-bit of arbitrary output signals, so you can build any digital system that uses combinational logic in PROM (and RAM for sequential logic), including a CPU. And since it's a PROM, you can reprogram it to implement another different logic device, simply by burning a new truth table.
After I realized this, the existence of reprogrammable hardware like FPGAs no longer sounds like magic to me anymore. From this, you can also see that computers with finite RAM and ROM is not a Turing machine, but a Finite State Machine.
http://www.racketboy.com/retro/mister-fpga-the-future-of-ret...
My current project is AR glasses. An FPGA is decoding a displayport signal and driving the display.
Because writing RTL is much more fun.
And if I happen to need a little CPU for some generic control operations, I just add a soft core to my design.
https://www.ecmwf.int/sites/default/files/gpsro_lecture_2015... is an overview.
There's rumors that a couple of the big cloud vendors use them on network cards for SDN.
Integration with non standard peripherals like ku band radios on a satellite.
No rumours needed, the papers are out there in the open.