What fully open integrated circuit of any kind can you buy? Maybe there exist simple 74xx chips with completely open designs, taped out using open source tools, manufactured using completely transparent processes, and then independently verified to prevent backdoors? I'm not aware of any, but I suppose the military have something like this.
Anyway depending on your level of trust at the moment you could:
* Download rocketchip (BSD licensed) and run it on an FPGA. You'd have to trust the FPGA vendor and their proprietary tools like Vivado. This lets you run Linux, at a speed of about 50 MHz.
* Run PicoRV32 on the reverse-engineered Lattice 8K FPGA using the open source toolchain. You cannot run Linux on this, but you can run short C programs without libc, and given that Clifford has done a very good job fully understanding the FPGA we can be reasonably sure there are no backdoors. My experiments with this are here: https://rwmj.wordpress.com/tag/icestorm/
