There seems to be a lot of misconceptions regarding the CA, and we are going to fix them, shall we?
- The CA
is not Superman: the CA generated eons ago when the white page (Primal Monitor) probed the stories (DCU) with a probe, its first and most powerful "son", Dax Novu.
The CA is the story and the idea behind Superman, shaped by the limitless powers of the white page itself (through its probe Dax Novu) in order to have the aspect of Superman himself. It's a doomsday weapon made of pure thought, and the story of Superman was chosen by Dax Novu because it's the most powerful and important story of all in the whole DCU.
As its name says (THOUGHT robot), the CA is pure thought which existed for eons at the verge of the white page, outside of space and time, waiting for the final battle against Dax Novu itself.
In order for the CA to "activate" and for Superman's mind to "control" it, the power of fusion of matter (Superman) and anti-matter (Ultraman) is necessary, and this is a kind of power which balances universes.
Quantum Superman (one of the most powerful versions of Superman out there) controlled the matter-antimatter reaction in the palm of his hands:
Spoiler: click to toggle
- The CA is bigger and more powerful than the Orrery of Worlds, which contains the whole DC Multiverse, and also it existed in the highest sphere of DC hierarchy, i.e. at the Source Wall, on the verge of the Overvoid; in fact, even planes of existence which are way higher than the mainstream Multiverse (such as the Heaven itself, the Word of the Voice (with the Voice being the Presence) where multiversal abstracts like the Spectre live) are completely irrelevant to it.
That's why, in the common pic posted by FutureProtagonist here above, you can see the Thought Robot looking at Limbo (which is as big as a whole universe and it's in the same sphere of existence of Heaven and Hell) like it's a small CD.
- The Thought Robot can adapt to anything, since it had to fight the original probe itself, corrupted by the dualities of stories and now known as Mandrakk Dax Novu. In fact, Dax Novu has limitless powers to begin with (since, as explained before, it's a part of the Primal Monitor itself), plus it fed on the Bleed for an infinite amount of time. The Bleed is the blood of the Multiverse, the essence of story itself, only one single drop possesses eternal power, and only the Monitors can manipulate it.
Dax Novu, before corruption, knew about all of this, that's why it shaped the CA with the ability to adapt to ANY threat; still, Mandrakk Dax Novu was so powerful that the Thought Robot, despite its victory, was damaged beyond repair.
- The CA and Dax Novu fought at the Source Wall, which is the verge of the white page (Primal Monitor), and their fight took place on a metaversal scale, since Dax Novu wanted to eat stories, while the CA's purpose was to protect them. What for the CA and Dax Novu are "stories", actually are life, space, time and existence inside the DCU itself.
- Mandrakk II Rox Ogama, another Monitor with an infinitesimal fraction of powers of the original Mandrakk Dax Novu (who fought the TR outside of existence) entered the DCU and stomped the Spectre and the Radiant, which, as well known, are two aspects of the Presence (respectively, vengeance and mercy).
Oh, and the Spectre was nearly at its best, since, in the same book (Final Crisis), it remade the whole creation (and remember that, form the CA point of view, creation was just a "story" to protect) freeing it from the Anti-Life Equation of Darkseid:
Spoiler: click to toggle
Now, I'm not here to say which "thing" would win between the TR and Yog-Sothoth, since the latter is something basically unimaginable in power as well.
My purpose was simply to clarify what the CA is and the levels of power it worked at.