Anything that would make a fun ASIC could be turned into an FPGA project. Something like Ben Eater's GPU project, which he did on a breadboard, could be implemented on them, or you could make a toy CPU with RISC-V, or create a softcore (faster than a software-simulated, slower than actual hardware) version of any number of older processors.
I have an FPGA and I've loaded up a sample that outputs a VGA signal[0] but the output is only based on hardwired patterns, not driving by a frame buffer or CPU. I'm not sure where to go next in order to use the memory chip on the dev board (I have the Digilent Arty [1] as well as a MiSTer FPGA setup) or have some sort of frame buffer which I can use with a CPU that is also running in the FPGA. I'm 90% through building Ben Eater's 8 bit cpu on breadboards (just have to add the control logic) and I did the 65C02 project of his just before that, so I'm definitely in the market for learning where to go next with it. My dream is to have a computer that I have completely design control over :)
Edit: Link to Ben's GPU project: https://eater.net/vga