These days, he's busy becoming a Maester in Oldtown. While there, he's reading ALL the restricted books, and learning ALL the things (like where the dragon glass is hidden, and even curing Jorah from his stone flesh disease - legend).
Now, it looks like a detail in episode 2 confirms a popular fan theory about him.
You know how in the opening credits of the show, it shows the entire map of Westeros, and that weird mechanical sun thing is hanging over it?
That same sun thing is hanging in the library at Oldtown.
A comment made between Sam and Archmaester Marwyn (aka Professor Slugorn) is what set it off.
In the scene, Marwyn tells Sam that a little "style" is needed in their stories, if people are going to actually read the histories they write.
Therefore the title he drops of, "A Chronicle of the Wars of Following the Death of King Robert I" won't do, so Sam suggests, "Mm, possibly something a bit more poetic?"
YOU KNOW, LIKE "A SONG OF FIRE AND ICE" (the actual name of the GoT books IRL).
It suggests that what we have seen, from season one episode one, to now, is a recollection of Sam's, written in his historical book.
IRL Sam, John Bradley even suggested in 2016, while talking to The Hollywood Reporter:
"If you take the logic of the story now, the story of Westeros and the story of the battle for the Iron Throne, it would be a book in that library.
"The visual motif of that is you're about to be told a story - the sense of an idea of being told a story, and people gaining that knowledge, the way Sam is absorbing knowledge in the library."
Why else would that mechanical sun thing be hanging over the map of Westeros and in the one place Sam is writing?!!?!?!?!??!?!?!?!