Here’s the important point I think you are missing. You are looking in TIME, not in space. When you look very far away, you are looking back to the beginning of time, back to a singularity where all matter and energy existed at a single point. In every “direction” you end up looking back to the same time and the same place, the beginning of everything. There’s nothing “beyond” that, it is the beginning.

If you ask, what exists today, beyond 13.7 billion light years from earth, the plausible answer is “mature galaxies like ours” but there is no possibility of collecting data or evidence of what is there, since the evidence would take more than 13.7 billion years to reach us, and in fact would never reach us because the expansion of the universe means that distances are getting larger between 2 points all the time.

But if we're looking in the direction of a galaxy far far away (i.e. 14bn ly away) (which we don't know it's there, but let's assume it is, because likely some galaxies must be there), how come we see the CMB, which is 13.7bn years old, and not the galaxy itself, which is (by definition) less than 13.7bn years old?

Because what we see from that direction are the things that were there in the last 13.7bn years. But the farther we look we see older images. So while we can see image from 10bn years ago of something that's 10bn light years away, when we try to see something 13.7bn light years away we just see microwave soup because that's what was there 13.7bn years ago. And if we try to see something 14bn light years away we see nothing from there because 14bn years ago there was nothing there.

Is it true that space had no volume at the Big Bang? I understand that density rises without (known?) bound as we approach the Big Bang, but even if you compress infinite space by an infinite factor you might still have infinite space -- infinity divided by infinity is undefined.

If you run the density backwards far enough, eventually everything in the visible universe is effectively at a single point, in some theories this was modeled as an actual infinite singularity, but it’s quite possibly unknowable since our first evidence is the CMB 380k years into the whole thing. As we rewind time from there things being to get highly speculative.

To "what exists today, beyond 13.7 billion light years", I'd rather say that "it doesn't exist yet". Because, otherwise, it assumes there's some "global universe time" like UTC or something. But all there is is our light cone.

That is fair. We are veering into metaphysics or religion if we talk about what exists there today, or even if there could be any relationship between our “today” and their “today”. By definition there is no causal connection. I’ll get back to you on that once I finish inventing my Time Machine.

The "observable" universe is about what we will ever see in the future, rather than about the past. If you waited around for billions of years, you could watch those early galaxies evolve.

There might be galaxies even further away, but you can't ever see them, not even in theory. The light from them will never reach us because they flying away from us further than the speed of light.

The CMB isn't really about galaxies. We know there won't be any galaxies past the CMB because galaxies couldn't have formed before the CMB was emitted.

In theory we can "see" past the CMB using gravitational waves. (There was a thought, in fact, that we'd already done that, but that appears to have been faulty.) The CMB is just kind of a practical limitation rather than a fundamental matter of spacetime: you can't see because it's too cloudy.

The question of whether galaxies beyond the observable universe "exist" is kind of a matter of metaphysics rather than astrophysics. As an astrophysicist, you basically just say they don't exist and you're done with it. But if you want to know where the universe "came from" (whatever that turns out to mean), you try playing around with notions like "our universe is an observable sub-part of a wider ensemble, which we'll never detect, but here's a pretty set of equations which explain our universe in terms of it".

Due to accelerating cosmic inflation, some galaxies are receding from us at faster than the speed of light, so their light will never reach us. The majority of galaxies (everything outside of our local cluster, IIRC) will eventually be receding faster than the speed of light and be beyond our vision forever.

Yes, this means most galaxies will appear to actually pass through/into the cosmic background, from our point of view.

I thought the logic of inflation had no limit, and eventually, everywhere, even atoms themselves would come apart.

I don't think we understand enough about the dark energy driving inflation to say that. If the force is constant for a given pair of particles at a constant distance, and the atoms are currently stable, that would seem to imply that the force from dark energy pushing the atoms apart will be the same as they are now.

However, as I said, I don't think we know enough about dark energy to say anything about its effects on the atomic scale either now or eons hence.

You're describing the Big Rip. It's now thought that the acceleration of expansion is not high enough for that to happen. Gravity is strong enough that the particles inside galaxy groups will stay causally connected indefinitely. Long enough into the future, whole groups will become disconnected from the rest of the universe, though.

Ah, I had never thought of it that way for some reason. Duh. Thanks.

No, because galaxies almost certainly exist which are further than the CMB. It's just that the light travelling from them has not been able to reach us in the time since it was emitted.

And never will reach us, as the space in between us and that distant light is expanding to quickly.

Well most physicists assume it is massively bigger than we can see. It is in principle possible that we see nearly all of it, but that would be very coincidental. The only thing we know is that it is at least as big as we can observe, but it could be hundreds of times bigger, 10^100, or even infinite.

There are speculations one could make that imply a minimum size, I recall a reading a prediction of 10^50 times bigger or so.