You can't as far as I know. Unless you want to get Ruby Ridged.

You have to do paperwork to get a title. One way to avoid having to use your social MIGHT be to set up a trust that the land will be purchased under instead of your name, but, you'd need to involve a really, really good title attorney to see if that's even feasible where you want to buy.

Also, you could probably find someone in Alaska that would "lease" you a parcel in the middle of nowhere for pretty cheap. You wouldn't own it, but if the lease is structured well, it's going to give you enough rights to do what you want on the land even if it isn't yours.