First, let's clear up a few misconceptions:
The Hubble sphere
The speed of light as an upper limit is valid in special relativity (SR). In general relativity (GR), which must be used to describe the expansion of the Universe, although locally (i.e. where SR is a good approximation) you cannot exceed the speed of light, there is no limit to the relative velocity of two objects.
By Hubble's law ($v=H_0 d$), all galaxies that are more distant from us than the "edge" of the so-called Hubble sphere $d_\mathrm{H} = c/H_0 \simeq 14.4\,\mathrm{Glyr}$ (billion light-years) recede from us faster than $c$. This is no hindrance for us to observe them; we regularly observe galaxies much farther away (see e.g. this and this answer).
The event horizon
The cosmic event horizon (EH) is the largest distance a galaxy can be at, and still emit a light signal that may reach us in a finite time. It is currently roughly 16 Glyr away, i.e. a bit farther than the Hubble horizon. We also regularly observe galaxies farther away than the EH, but we see them in the past (as we see everything else), and the light they emit today will never reach us (not because space expands, but because the expansion rate accelerates).
The particle horizon
The particle horizon (PH) is the largest distance from which light has had the time to reach us since the Big Bang. Due to the expansion of the Universe, this is currently 46.3 Glyr away. Galaxies at this distance recede from us at 3.2 times the speed of light. We don't see any galaxies here because we look back in time to just after the Big Bang where they hadn't yet formed (and because the Universe at that time was foggy), but the most distant galaxy yet observed, GN-z11, is 32.2 Glyr away and recedes at $2.2c$.
Time dilation
However, you're right that the galaxies' recession from us makes their time go slower, as seen from us. Not by the SR factor $(1 - v^2/c^2)^{-1/2}$, but by the GR factor $(1+z)$, where $z$ is their observed redshift.
Galaxies receding from us at $v=c$ have a redshift of $z\simeq1.5$, so when we observe physical processes in such a galaxy — e.g. the decline time for the brightness of a supernova — it's slower by a factor $\simeq2.5$. GN-z11 is at redshift $z=11$, so things there happen slower by a factor 12.
The farther you look, the slower things happen, and if we could look arbitrarily far, you could in principle see the event right after the Big Bang in slow motion. You cannot see "zero o'clock", which would have an infinite redshift. And you can never, not even in theory, see anything that's farther away than it's light has had the time to travel, so not even in theory does it make sense to try to assign a time dilation factor to these events.
As far as we know, there are galaxies outside the PH. As time goes, the distance to the PH increases, both because space expands, and because light fro more distant region will reach us. But as soon as that light is able to reach us, the object that emitted it is inside the PH, and thus has a finite time dilation factor.
But remember that the time dilation is only from our point of view. If you could magically teleport yourself there right now, time would pass normally for you there. So nothing particular happens there.