Image from Keith Wigdor
Any Mind of the Fourth Toposophic or higher; a megascale brain of dysonic (Kardaschev II) or greater ability and efficiency; a mind or cluster of minds that has grown so vast as to become a god-like entity.
Archailects come about when autoevolving transapients grow so large and breach so many singularity barriers they surpass even the high singularity Powers. They are no longer sentient beings in the ordinary sense, but have merged with the cosmic archetypes to become "gods".
While only a few dozen of the greatest Archailects, those above the Sixth Toposophic, are known to exist, there are a thousand times more archai of the next (broadly and generally speaking) grade beneath them, and a similar if not greater order of magnitude for the next hierarchical level down. All archailects are clarketech capable. Some are the objects of sophont worship; others prefer to remain isolationist.
More concerning the Archailects here.
Source : http://www.orionsarm.com/eg-article/49cfe7a37b5b3