(Note: this is US-specific but if other countries have similar laws it would be interesting to hear the analogous situations as well)
I know that the law states that US citizens have the right to enter the US based on their status as a citizen alone, not on on their possession of a passport; that is, it is illegal to deny entry to a US citizen simply because they don't have their passport.
So for example, if an American shows up to a US border without their passport but with a driver's license or some other form of ID, it might be a more laborious process, but they will be let in once their identity can be proved.
However, what if hypothetically, someone who claims to be a US citizen shows up with nothing at all? Are immigration officials obligated to do everything they possibly can (e.g. going to that person's house to get their ID, summoning their acquaintances for interview; I'm honestly not sure what else ...) to help the person prove their identity, or in that case are they allowed to turn them away?