On the other hand, demons generally don't leave ACs behind, so your question seems like the kind of thing a demonologist would be interested into.
[look at you quesitorially]
Well... if we are talking about a person, I think one who never stays in one place for much time and one who has no personal belongings will have no AC, besides what one's body leaves behind. This might not be that hard to achieve for the trully destitute, or for the filthy rich (who can afford to discard things after using them once or twice).
Maybe if you are very detached from mundane life you'd cease to leave connections? I'd look at the eremites and ascetic monks, or at the Criamon with the highest attainment on the Enigma.
People leave an absurd amount of arcane connections- write a letter? AC, cut some hair? AC. Sweat? AC.
Daemons, demons, spirits in general have true names as ACs, so they pretty much have them by definition.
Inanimate objects are pretty much the only things which have a chance of having an AC, and even then they would need to be fairly simple (an arrow fired from a bow is an AC to the bow, ink from a pen an AC to that pen) and tough (brick dust from a brick wall? AC. Rust from a metal object? AC. Splinter from a wooden object? again an AC)
celestial bodies do admittedly qualify (so far as anyone knows, when you can't penetrate the lunar sphere the question of moon rocks are rather irrelevant)
Just nitpicking: many lesser magical spirits may well have no True Name. See RoP:M, p.110:
Only creatures with a Might score have a True Name, but not all creatures with a
Might score do. All demons, angels, and Daimons have a True Name, as do other
intelligent entities with a unique personality; the storyguide is the ultimate arbiter as
to which creatures have a True Name and which do not.
Lesser spirits do sometimes have "normal" names and those do work as ACs. Perhaps more crucially a spirit is generally the spirit of something, and that something (or its physical representation if it's abstract - e.g. an angry person for the spirit of that anger) is an AC to the spirit. So yes, despite the aforementioned nitpicking, it's hard for spiritual entities not to have an AC at all.
The big key line from the book is Mystically, the Arcane Connection is still a part of the target, thus making the spell possible.
So, yes. Your shed hair is a part of you, your composed and handwritten note is a part of you for a time, the hammer you use every day for the last few years is part of you.
But that quill you picked up from the senescal to write the note is not. The letter you dictated to someone else is not.
As stated, it is all one-way. I think Hermetic magic could have a spell that targets your vision, and illuminates any bodily-based ACs you've left behind, but I don't think it can cast a wide-range scrying spell to find all the ones from the last month in a city.
I could see perhaps an In spell for each form "finding the missing parts" on the basis that while the body is not an AC to the hair if thr hair is mystically a part of the body then an In spell cast on the body should be able to locate it. Two caveats to this if allowed are that it limits to detecting ACs of a certain type- InCo may locate missing body parts but won't locate distant thoughts (letters, books, probably favorite items) which would require InMe. The second is that because the whole is not an AC to the part you would not have an AC range to work with. Being In you could use sight or sound to locate nearby arcane connections through this method, but something 50 miles away you will never notice.
Objective #1 is just to learn that a part is missing
Objective #2 would be to know if the missing part is "Degrading Naturally and on its way to no longer being considered as part of"
I think this discussion has concluded that both are possible without breaking the AC limit.
To locate the missing part is a bit more tricky as we do not have an AC to the part. I think we call all agree that a container spell (Circle, Room, Struture, Boundary) would be able to locate the missing part if within said container. Going beyond that would probably break the AC limit.
Number one could be achieved as simply information about the object itself, to a degree. If I cast an InHe on the ship of Theseus I can detect what parts have been removed, or removed and replaced with new parts, up to some limit based on the magnitude of the spell (I would presume) because these are properties of the ship of Theseus.
If the captain's wheel was removed from the ship of Theseus then I cannot find any information about that captain's wheel from a spell cast on the ship of Theseus. I cannot know if the wheel is decaying, is preserved, or if it has been burned for firewood or carved into something else. Note the key point of a spell cast on the ship itself. If I have a room target spell that will let me know if any wooden objects in the room are not decaying that would be perfectly legitimate. For that matter I can use an InVi to detect all AC's in the room to a particular object (I would expect a spell detecting all AC's in general would produce an overwhelming amount of information)