What is the subjective inverse of "invite" or "invitation"?

If you are about to do something and you ask someone to join you, you "invite" (verb) them, or send them an "invitation" (noun). But, in the opposite direction, if they ask to join you, what verb or noun describes their action toward you?

I believe the best word for to invite oneself is the verb, to impose, or its noun form, imposition. While the exact meaning regarding invitation is slightly ambiguous in this usage, it does describe their hindering behavior well. The second definition of impose from the Cambridge English Dictionary comes rather close to your desired purpose.

To cause inconvenience to someone or to try to spend time with them when they do not want you to

Depending on the level of formality, you could also use the colloquial adjective self-invited. However, if this is for anything more than an informal exchange, I would strongly encourage the usage of impose.