Why "mich"?
Ich kann mich daran gar nicht sattsehen.
Instead of,
Ich kann mir daran gar nicht sattsehen.
The thing I'm looking at is something meant by "daran", so doesn't accusative "mich" make it sound like "I'm looking at myself"?
My confusion comes from a similar construct,
mir etwas ansehen
Don't concentrate that much on sehen, rather on the meaning of satt when trying to make sense of sattsehen. That's crucial for understanding why the accusative is correct here.
The Duden gives the following meaning for sattsehen:
sich etwas so oft, so lange ansehen, dass es einem reicht
Langenscheidt gives the following possible translation:
Not to be able to see enough of
What you are looking at is daran.
And your action (sattsehen) does have an effect on someone, namely you. Hence it's accusative (Wen oder Was? -> mich. Wen sehe ich satt? Mich!).
You cannot look at daran for so long, that you have enough of doing so. You are not able to see enough of daran.
Satt can mean full up, satisfied, satiated. It's usually used when you ate enough and now cannot eat more. (as guidot noted in a comment, sattessen has the same accusative construction).
In your example, it's a bit complicated because it's a kind of double negation:
First, note the formal difference: etwas ansehen already has an accusative object. Therefore, if another nominal object is to appear, it must be in the dative. (There is only a handful of exceptions to this rule.)
Second, the pattern that builds sattsehen is productive. The adjective is interpreted as a resultative predicative pertaining to the single object, which is in the accusative. The accusative is to be expected in that the object is undergoing a change of state, with the resultant state being described by the adjective. In the case of sich sattsehen, think seeing so much of something that you have become satiated (i.e. don't want to look at it any further). Further examples:
Ich habe mich müde gelaufen.
"I walked so much that I am tired now."Er hat sich schwarzgeärgert.
Lit. "He was so angry that he is now black.", i.e. "He got really angry."Sie hat sich warmgeredet.
Lit. "She has talked so much that she is now warmed-up."Man hat ihn grün und blau geschlagen.
"They hit him until he was black and blue."Es hat schon wieder jemand eine Katze totgefahren.
Lit. "Someone drove a cat dead again.", i.e. "Someone ran over a cat again."
The predicative doesn't need to be an adjective.
Sie hat sich in Fahrt/Rage geredet. (similar to the third example above)
Note that whether the object is reflexive or not is dictated by plausibility. So although sattsehen is always reflexive, müde laufen does not need to be:
Die Mannschaft hat den Gegner müde gelaufen.
"The team ran their opponent ragged."
Sattsehen likes to appear in the idiom sich (an etwas) nicht sattsehen können.
Die Landschaft war so schön, daß man sich an ihr nicht sattsehen konnte.
"The landscape was so beautiful that you couldn't get enough of it."
The "mich" part just refers to the fact that one is the actor himself e.g. you are the one who does the looking. Compare to the term "sich erinnern" (to remember) -> "ich erinnere mich" does not mean 'I remember me', but rather that I am the one remembering something/someone
So in the case of "mich sattsehen" the mich is not ruled by the act of looking, but by the sating of the desire to look at someting :) As such it roughly transcribes as 'I sate myself (accusative) by looking at sth.'
I think your confusion is about daran. It's very different from etwas, because it replaces a prepositional object or adverbial. That's because of the built-in preposition an:
Ich kann mich daran gar nicht sattsehen.
Ich kann mich an (so) etwas gar nicht sattsehen.
In contrary, a plain etwas is indeed an accusative object.
"Ich kann mir etwas gar nicht sattsehen."
Ich kann mir etwas gar nicht ansehen.
In contrary to ansehen, the accusative object of sattsehen isn't the object ogled at however, but the thing or person which experiences the effect. That's the common use for the accusative object.
Ich kann mich gar nicht sattsehen.
This is similar with German verbs which take two accusative objects
Sie lehrt mich etwas Neues.
What you have there is a classical German construct of sich + verb. That's the reason why it conjugates the way you describe.
Take for example sich + freuen (to be happy):
The forms are:
Similarly, in your sentence the construct is sich + sattsehen (to be satisfied watching). However, the Grammar slightly changes because sattsehen is a separable verb. Therefore:
This can be further extended by the form sich an etwas/jemand sattsehen (to be satisfied watching something/someone). With this, you are able to specify what you are watching. For example:
However, for a person this might be:
Or for a narcissist:
The above might be the the reason for your confusion. These sentences look quite similar, but are in fact very different in meaning.
Last but not least, you have a negation with können in your sentence:
This changes the Grammar to the way können works. (sattsehen is no longer separated, as it is not the predicate in this sentence.) Furthermore, the meaning switches from: I'm satisfied watching [...]. to I cannot stop watching [...] (satisfiedly).
A nice way to look at it is that "sattsehen" doesn't so much come from "sehen" but is more related to "gewöhnen". When you get used to the sight of something extraordinary, you can say this: "Ich habe mich daran gewöhnt, es zu sehen." Or you can say the following: "Ich habe mich daran sattgesehen."
You probably shouldn't look for logical consistency in any human language, after all, we've failed to describe them mathematically, no matter which brilliant minds tried just that. It's the same in English. The only thing we can do is make it easier for ourselves to remember the quirks.
