There are many theories of cognition. But if you want to work within a framework with the following properties:

• Explains the major cognitive phenomena we know about.
• Fits experimental data well, down to human reaction times, in a wide variety of psychological experiments.
• Has a relatively complete story about functional neuroanatomy.

Well, then, I'm not aware of any theories which fit the bill as well as ACT-R theory. 

You might also be interested in the common model of cognition (initially named standard model of the mind), which is consistent with the ACT-R picture, but also consistent with several competing theories. Think of it as the high-certainty subset of ACT-R.

References for ACT-R

I am no expert on ACT-R, so unfortunatery I can't tell you the best place to get started! However, here are some references.

Books by John R Anderson

John R Anderson is the primary researcher behind ACT-R theory. I have not read all of the following.

• Learning and memory: an integrated approach. This book is not about ACT-R. It's basically a well-written textbook on everything we know about learning and memory. I think of it as John Anderson's account of all the empirical psychological phenomena which he would like to explain with ACT-R.
• How can the human mind occur in the physical universe? This is a book about ACT-R. In this book, John R Anderson seeks to spell out "how the gears clank and how the pistons go and all the rest of that detail" -- which is to say, the inner workings of the brain.

John R Anderson also has several other books, which I haven't looked at very much; so maybe a different one is actually a better starting place.

Other References

Here are some other potentially useful starting places.

References for the Common Model of Cognition

• A Standard Model of the Mind: Toward a Common Computational Framework across Artificial Intelligence, Cognitive Science, Neuroscience, and Robotics. This is the original paper for the common model of cognition. It rests on the observation that many cognitive architectures seem to be converging to common principles, much more than they were even 10 years earlier. To codify this consensus, leading researchers behind three of these theories got together and made a close comparison. (One of the three is my phd advisor, which is how I became aware of this.)
• Empirical Evidence from Neuroimaging Data for a Standard Model of the Mind. What it says on the tin.

If you search "common model of cognition" in Google Scholar, you will find a number of other papers discussing it.

How should we evaluate all this?

• It doesn't scratch the same itch that Solomonoff Induction and other ideas more commonly discussed in this community do. Like, at all. The ACT-R interpretation of the question "how can the human mind occur in the physical universe?" has little to do with embedded agency questions, and much more to do with computational psychology.
• Although it is supposed to be a computational model of cognition, ACT-R is not an AGI. I would say it's not even a proto-AGI. The ACT-R community cares about closely matching human performance in psychological tests. It turns out this task is a lot different from, say, building a high-performance machine learning system. ACT-R is definitely not the latter.
  • As such, despite ACT-R's success at replicating human learning behavior in the context of psychology tests, we might broadly conclude that it's missing something which modern machine learning is not missing.
• I'm also not claiming that you should believe ACT-R. If you take existing theories of cognition and compare them in Bayesian terms, I think ACT-R comes out on top. But ACT-R has a lot of pieces, and I don't think all of them are necessarily correct. And a lot of the reason ACT-R comes out on top in evidential terms is that it makes concrete predictions (which fit the data). Many other theories of cognition just aren't at that point, mostly because they don't focus on that so much. But you could plausibly think ACT-R is overfitting by including a new mechanism every time it needs to to fit the data.
  • For example, I would say that Solomonoff Induction is in some sense a theory of cognition; as is AIXI. But, clearly, neither are trying very hard to fit data from experimental psychology.
• Nonetheless, I'd wager your personal theory of functional neuroanatomy based on ideas from modern machine learning and/or bayesianism is probably worse overall, at least in terms of fitting with experimental data, which is ACT-R's bread and butter. So it might be useful to study ACT-R, if you're into this kind of thing. It might at least try to explain some things which you hadn't tried for. And it might even have some good ideas about how to do so.

I don't personally think about ACT-R very much, but that's because my thinking on AI alignment has little to do with neuroanatomy-inspired AI. Some other people around here think a lot more about those things. ACT-R theory might be useful to those people? Also, if you care about the nitty gritty of human modeling, EG for the sake of inverse reinforcement learning or other value-learning purposes, ACT-R might be useful. It is, after all, a really sophisticated model of a human.

Personally, I am hoping that learning more about ACT-R theory could help me think about human (ir)rationality in more detail.