Can a list of hashes with property $X_{n+1} = \operatorname{SHA-256}(X_n)$ loop? This is the situation:
I have a list of hashes where each hash is always the SHA-256 of the one above it.

If I keep going down the list it must at some point loop since there can only be $16^{64}$ possible SHA-256 hashes.
So at some point the hash I get has already been in the list before.

The problem is that the algorithms behind SHA-256 is above what I can currently comprehend (I'm only 17 and a part free programmer for 3 years).
But on here there seem to be a lot of geniuses so my question is:

Is SHA-256 designed to loop after exactly $16^{64}$ hashed hashes, or is there a chance it will do so before that point?

Also: If so, it would probably do multiple times.

I saw the post on $\operatorname{Hash}(x)=x$ with $p=1-\frac1e$ answered by User fgrieu, saying that we don't yet know. But if this is determined by probability is my question also answered by probability? Please let me know if I'm thinking the wrong way or give me an answer if you have one.
Is SHA-256 designed to loop after exactly $16^{64}$ hashed hashes, or is there a chance it will do so before that point?


SHA-256 behaves more or less like a random function. You can expect it to loop after approximately $\sqrt{2^{256}} = 2^{128}$ iterations due to the birthday paradox