I'm imagining someone importuning another person. For example, if someone asked another person if he or she could come join them on an outing, and when the other person doesn't seem too keen on the idea, the person who asked the question could say, "Please?" Using an exclamation mark gives the sense that the person is raising his or her voice, and a period just doesn't seem strong enough at all. Could a question mark work here?
The question mark at the end of a sentence, when quoting a speaker, causes me to hear them saying it with a rising intonation, as if they are pleading.
An exclamation mark makes it seem like the speaker is insistant, raising their voice or shouting.
If they are importuning (harassing persistently), I would use "Please!"
Take "The sky is blue?". "The sky is blue" is an expression of fact, but obviously I am asking if the sky is blue, although gramatically nothing about that is a question except for the presence of a question mark. The presence of a question mark implies I actually want confirmation of this fact.
You can apply the same logic to requests and commands. Take "Give me that pencil". If you add a question mark to make "Give me that pencil?" (strange when written, but fine spoken), it implies that you want confirmation that your request/command has been received. "Please" acts as an expression of politeness that can be added to requests, so we can be more polite by saying "Give me that pencil, please" (the same meaning as "Give me that pencil", but more polite) and "Give me that pencil, please?" (the same meaning as "Give me that pencil?" but more polite, and this time maintaining the request for confirmation).
The final step would be that, if the meaning is clear, you could emit the actual body of the question if the meaning is clear. If you were pointing to the pencil at the time you could just go "Please?" and you will be understood. Of course, without physical context, the use of "Please?" on its own would usually mean "Can you repeat that?". Nevertheless this isn't common in English but is very common in, for example, German, where "Bitte?" would be unambiguous in this meaning. Likely the reason that "Please?" despite being uncommon English, would still be understood, is because of the Germanic roots of English (among other roots).