I remember reading a word once which had this meaning which starts with the letter 'a', like "agritocracy" or "autotocracy". I just can't remember the word now; it specifically means rule of a society by an artificial general intelligence. Mostly I'm creating this question for self-tracking. I already know such a word exists because I read it and looked up the meaning in the past. I just can't remember the word a few months later.
According to Wikipedia, "automata" (the plural of automaton) is a theoretical branch of computer science that has played a major role in artificial intelligence. The word you are looking for, therefore, would be "Automatocracy". See Wikipedia link
Though the following word doesn't begin with the letter 'a', Wiktionary lists
rule by robots
The robots involved would obviously have to possess intelligence at some level; [Wikipedia] licenses this (though the very existence of the word virtually demands it) here:
Robots can be autonomous
And in this article:
Autonomous robotics is usually considered to be a subfield of artificial intelligence, robotics, and information engineering
though of course outward appearance need not be humanoid.
The term automatocracy is mentioned (with the equivalent of scare quotes) in this Wikipedia article on 'Political ideas in Science Fiction', but has a very limited circulation. Though it has been used outside the SF community; this Reddit article on politics uses the term. It is perhaps borderline acceptable, needing scare quotes outside SF at the moment.
The word I was trying to recall is algocracy. I realize this isn't so much rule by AGI, but moreso society where law is executed as code, which could include an AI program, but might also include blockchain smart-contract-based legal systems.
I know this must be the word because I was ran into this term in a transhumanist forum originally, so this adds up.
I do like people's answer automatocracy though; that seems like a much better word to describe rule by AGI specifically.