The far limit of the visible universe is the cosmic microwave background, which is the heavily red-shifted view of when all of space was filled with an opaque plasma of similar temperature to the surface of a star[0].
The process that caused the light we see as the CMB happened everywhere in the universe roughly all at once. At the time, from any given point, you'd just have seen blinding light but as things cooled, you'd see a sphere expanding around you where this light was dimming, as less and less, and finally none was being emitted around you, but due to the finite speed of light, light from outside that sphere would still continue arriving in your eyeballs. That's the situation we're in today; that sphere's just really big now.
During recombination epoch ~400,000 years after the Big Bang, this light would have been visible. Due to expansion, over time that light has stretched to longer and longer wavelengths, and we currently see it as microwaves.
Note: It's been ~30 years since I was actually studying physics & astronomy; others may be able to offer better explanations or correct me.
The CMB happened everywhere in the universe at the same time[0], so what we're seeing is the light which took 13-point-whatever billion years to get here[1] from some part of the universe in that direction at that younger age.
> My understanding is that the early galaxies still produced light after recombination.
I'm not sure what you're imagining about recombination, because I've not heard any suggestion of any galaxies existing before it, so they only produced light after recombination.
[0] for some definition of the concept, even though relativity is formulated with the assumption that there isn't any good concept of simultaneousness.
[1] from our point of view. From light's point of view, time isn't defined.
The galaxies we're seeing are in front of that.
[0] 3000 K, https://en.wikipedia.org/wiki/Recombination_(cosmology)