If you are pinging someone who’s busier than you, you should have a single well-thought-out specific ask. Only ask for something you truly cannot do on your own, and where you have done everything you possibly can by yourself.
For example, if you’re asking someone to send an email on your behalf, you should have a draft message ready. If you’re asking them to sign something, you should prepare a docusign beforehand with the signature blocks indicated. Or if you’re asking for an appointment, you should do it at their convenience.
Asking for time is like asking for capital, so be efficient with your asks.