One mostly hears that the Andromeda Galaxy (M31), at 2.25 million
(2,250,000) light years is the most distant naked eye object. I've
seen it, and it is the farthest naked eye object I've seen. Oh,
alright, I was wearing contacts. But without them, or glasses, the
most distant object I can see is the Sun. For that, I have to wait
until morning. M31 is the largest galaxy in the local group.
The Triangulum Spiral (M33) is the third largest member of the Local Group, and some people claim to have seen it with the naked eye. It is a face on spiral galaxy. M33 is more like 2.78 million (2,780,000) light years.
Bode's Galaxy (M81), at 12 million (12,000,000) light years has been spotted by several people. This page at SEDS on M81 has a description of how to see it.
The trouble is, at Magnitude 6.9, M81 is dimmer than most consider naked eye. It depends on whose eye it is, and also where the feet are standing. It has to be an exceptionally dark sky site, probably at some altitude, at the right time of year, etc. Still, it's my answer to the question. I'm done. Finished. Except for the following speculation.
Apparently, SN1987A was a naked eye event. This was the big super nova in the Large Magellanic Cloud (LMC), visible from the southern hemisphere in early 1987. It wasn't a distance record holder, as the LMC is considered a satellite galaxy of the Milky Way. The LMC is only about 100,000 light years out. This is about the distance to the outer edge of the far side of the Milky Way. So one could consider the LMC to be part of the Milky Way. The LMC is already in the process of being torn apart, eaten really, by the Milky Way. The Milky Way is the second largest galaxy in the Local Group.
Since super novas can outshine their host galaxies, it seems possible that one could push the farthest object out a bit, if only for a few days. Say there is a galaxy, oh, about 15 million (15,000,000) light years away. It isn't a naked eye object now. But what if it hosts a bright super nova? That object would be the same distance, and could be naked eye visible for a few days.
There are other bright things too. I'm not aware of any naked eye visible quasars, for example. There is at least one quasar, 3C 273, visible in a 6 inch (150 mm) telescope. It's magnitude is 12.9. It sits about 2 billion (2,000,000,000) light years away. While no one would call this naked eye observing, it does come under the concept of eyeball astronomy - which is distinguished from astrophotography.
How far could one see with eyeball astronomy?
That, of course, depends on the telescope. It mostly depends on the light collecting area of the big end of the telescope. There is a story that several people have looked through one of the 10 meter (32.8 foot) Keck telescopes. Each of the twin Keck scopes were, at the time, the largest instruments on Earth. This pushes the concept of eyeball astronomy to interesting limits. Extrapolating from smaller scopes at good dark sky sites, one should be able to see objects as faint as 23rd magnitude with a Keck. Quasars that would appear around that magnitude might be far enough away to be red shifted out of the visible spectrum. But, perhaps higher frequency photons, maybe X-rays, will have been red shifted into the visible spectrum. It seems reasonable to assume that one would be able to peer most of the way back to the beginning of the Universe, or most of the way across the visible part of the Universe. At these distances, due to the expansion of the Universe, light years are not even approximately years back in time. The Universe is about 13.7 billion (13,700,000,000) years old. However, the distance to the most distant detectable objects is about 46 billion (46,000,000,000) light-years.