- Pre-screening
- Making travel arrangements based on known preferences
- Setting up meetings with other people who are hard to reach and also have busy schedules
- Following up on various things
And so forth. Not that I'm at that level but my understanding is that executive assistants can be incredibly helpful.
If you can afford it, you can start doing it before you get too busy. In fact you can do it as a means of staying less busy (at least work/chore-wise).
(Even when I was a summer intern, I shared an admin who gave me leverage: she played phone tag, tracked down pricing and availability from vendors, and told everyone who was not explicitly whitelisted that I was "in a meeting." But we also had offices with doors back then too, so maybe it was a different world?)