A bit funny that you would think the typical government was modern enough to allow you to interact with them through cryptographical proof ;)
That said, there is AFAIK one such country: Estonia. It's not anonymous or anything like that, but they allow you to get a ~~"e-citizenship"~~ "e-residency" (anyone can do it globally, I did it in the past, it is quite easy) which is basically a chip-card with a cryptographical proof of your identity. With it you can then interact with the government and banks etc. and start a business or a non-profit company etc. As Estonia is a EU member, it gives your company quite general access to any EU country, so it is quite handy to have in some cases.