Well... "cheap" is relative, but the Ice40 boards from Lattice are reasonably priced IMO. And more to the point, there is a complete Open Source toolchain available for programming them. That's a big plus in my book.

https://www.digikey.com/product-detail/en/lattice-semiconduc...

http://www.clifford.at/icestorm/

My standard to-go-to FPGA is an Cyclone II EP2C5T144 development board. On a good day, you can find them for less than $5 on eBay.

You'll need a USB blaster to program them, which adds another $10 or so.

They are small, but they are still large enough to add a little CPU with a little RAM, plenty of hardware multipliers etc.

Pretty much anything from Digilent. I've got an old Spartan6 board that has a boatload of horsepower. If you want to get fancy, look at the Zedboard family -- FPGA wrapped around ARM cores.

I also recommend Digilent, but they are not bargain basement. They're education targeted, often focused on the lab, rather than non-hobbyist R&D boards, but not $10.

I always recommend the DE0-Nano. It's getting a little old, but it's basically all right there, open for use. I'm a fan of the Arrow DECA which has a complete assortment of neat odds and ends. Lastly Digilent's Arty and CMOD series are awesome depending on what you need.

I know this makes a few assumptions on what you mean by "Cheap", but I find sub $200 for FPGA boards to be pretty affordable when compared to the gamut.

There is a large list here (including this one): https://www.joelw.id.au/FPGA/CheapFPGADevelopmentBoards The ICE40 chips are the ones supported by the open source icestorm toolchain.