[anyfoo]

Interfacing with arbitrary hardware, for example random LCD devices with sometimes proprietary on the wire protocols. (Don't have a graphics chip for that screen? Make yourself a graphics chip for that screen!) Or Digital Signal Processing.

[alexhutcheson]

If you get a high-end microcontroller like a Cortex-M, you can normally just bit-bang the interfaces to devices that you don't have hardware for. DSP is similar - for a hobby project you're normally better off using a high-end device like a Cortex-A and doing the processing in software, rather than futzing with custom digital logic in an FPGA. If you're on a power budget (e.g. for a portable device) you could use a DSP chip like the C6000, but writing the software for it is pretty complicated if you haven't used one before.

If you're building a device that's going to be mass-produced and sold, then the situation is different and using FPGAs can make sense, because you'll amortize the engineering cost for the digital logic across all the units you sell. It can be worth it if it lets you use a cheaper processor or microcontroller.