Sounds suspiciously like the United Kingdom. I don't think a single government service remains that cannot be done online. Not only that, but they actively invest a vast amount of resources into convincing anyone who tries to use physical services (mail, phone) that they are making a huge mistake. Anyone unfortunate enough to have to phone up any government office must to spend at least 3 minutes hearing about how much easier it would be to just use the website.
As a UK national overseas I don't know pains of calling up anywhere, and for sure there will be pains for many in varied forms, but find gov.uk consistently excellent from personal affairs to running a business. While not applicable as gov.uk does everything I need, surely channels exist for those than cannot, or unable, to use it?
Example: setting up national insurance contributions from abroad. You can't do that online, and so being repeatedly told to use the website while on hold on HMRC's phone lines (if you're lucky enough that their voice recognition system didn't hang up on you) is adding insult to injury.
