For anyone who just wants a taste of this I highly recommend the game Shenzen I/O. The game is your writing a subset of assembly for products at a manufacturing facility.

Great recommendation, although the limited assembly language doesn't give you access to memory or a stack. Zachtronics have a couple more assembly-language-like games - TIS-100 and ExaPunks - both of which are also excellent for exploring the kind of coding you can do in a very limited programming environment. ExaPunks offers the most programming freedom (including a fully emulated handheld video game device you can use to implement actual games); TIS-100 is probably most like real assembly programming.