What has been an amateur science breakthrough in the last century or two which didn't have at it's base some billions of dollars of government funding.
That was over a century ago (although within the “century or two” limit specified). But it’s not at all characteristic of the majority of scientific discoveries made in the last century, which were largely made by professional scientists and grad students on their way to becoming professionals. The list of Nobel Prizes in various sciences over the past century I think demonstrates this.
What has been an amateur science breakthrough in the last century or two which didn't have at it's base some billions of dollars of government funding.