'By any chance' adds a gentle tone of uncertainty and is less direct than just asking a question.

Subtle Uncertainty

It implies that you don't expect the person to know the answer but you're asking just in case.

Low Expectations

Using it at the start of a question is polite, especially when interrupting or speaking to a stranger.

'By any chance' makes a request sound more tentative and less demanding.

It suggests that you're asking for a favor, and it recognizes that the person might say no.

It can convey humility by acknowledging the possibility of the request being inconvenient.