why does my new iphone 16 pro max still say completing setup in background after a week of ownership?

As Bob states, these post-setup background tasks, mainly re-indexing all of your data, can take longer than a week. It ultimately depends on how much data you have. But you can speed up these processes by keeping your iPhone on a strong Wi-Fi connection and plugged in. If it doesn’t meet one or both of those it won’t perform much indexing.

Rnsimon1965 wrote:

How long will it take if I do that?

The answer is a lot quicker. As I said above the iPhone doesn’t do much if not any indexing when the iPhone is not on a strong Wi-Fi connection and a charger, so start by doing that, which usually is performed overnight. If you haven’t been doing that indexing will take much longer.

The message means what it says. It can take a long time for everything to complete. Give it a few more days.

If you will do it, means it’ll take less of your time than expected if you won’t,

again consider your iCloud storage terms and its availability as the more data you have you will need to upgrade your storage package

Are you saying that at this time, you don't have your phone connected to a strong Wi-Fi signal and the phone is not connected to the power adapter? How long have things been like this?