What is the maximum size of a Library, in TB, that the Photos app can manage in a Mac Mini?

I think that even though, as Yer_Man says, they're not limited, very large Photos Libraries can get a bit cumbersome. Some processes will slow down as a Library grows. There's no special problem in having several smaller libraries. I have a number of smaller libraries loaded with pictures I'm not likely to want to see at the same time. I also have a library just for the "favorites" that I share with my phone and iPad. While you can just double click on a library to swap them, I use PowerPhotos to help switch between libraries and to search through all the libraries together.

There is no published limit to the library size forPhots on Ventura.

For the previous versions of Photos, up to macOS 12 Monterey, we could infer from the fact that Photos has been supposed to be able to migrate Aperture Libraries, that Photos should be able to handle libraries with one million media files and 10000 albums. Photos can no longer migrate Aperture Library, so that clue is no longer valid.

But there are practical reasons to keep the libraries small. It will depend on your patience, how large you will let your library grow.

One big difference between Photos and aperture is the huge amount of background processing Photos is using to prepare the library for searches, also the syncing with iCloud.

• When you are using iCloud Photos, the device with the least storage will be the bottle neck for the library size.
• The background processes to classify your photos according to categories (faces, objects, scene types) are needing a lot of processing power to scan all photos, and not only once, but after each major system upgrade again, or after you modify the photos. And for some of them the amount of computation time does not seem to encrease linearly with the number of photos, but with a higher order, if photos has to compare photos for the duplicate detection or to compose layouts.
• Photos is also needing free storage and memory for temporary data, depending on the library size, to upgrade libraries to a new system version or when importing on library into another. Photos may simply crash, when trying to import a very large library.

i would try to keep the Photos Libraries as small as possible - I am currently struggling to keep my system photos library, that is syncing with iCloud Photos to all my devices ,not larger than 50000 photos and videos.This will still allow my to keep the favorites from all years in the library, plus the new photos that still need editing and tagging. when the library is getting larger, managing the library (upgrading to new versions, syncing again with iCloud, scanning again for faces, etc .) will be getting very slow. Also smart albums will be getting unresponsive, or simple operations like deleting a photo from the library.

How many items in the library? How much Ram do you have on the Mac?

Photos is a database and that means the db file is tracking entries - and the entries in the db are the same regardless if the file is 100kb or 100mb - so apart from Disk Space, the size doesn't matter. But to have a library that size you must have a very large number of entries, and that could be swamping the Ram.

There is no such limit, except the disk space available to you.