While there aren't any one bit CPUs, there are a couple of dozen CPU designs at
One bit CPUs are largely pointless. First, you're almost certainly talking about a microcontroller architecture, so only data memory and registers are actually one bit wide, and the trade off for program size vs. the (few) gates saved in the ALU will be really bad for almost all applications. There are a few uses of four-bit microcontrollers but only where the very small cost differential from an eight-bit can be collected over a very large number of units made, and then they're only applicable to a rather small number of applications.
However, a simple RISC or eight-bitter is well within the capabilities of someone with some background designing stuff ASICs or using the larger programmable logic parts, and with a few free months on their hands. You will quickly discover that the problem is not getting a CPU to execute a few opcodes, but building a software tool-chain so that you can do anything interesting with it.