Have there been successful attempts at spotting Neptune with the naked eye?
As pointed out here, the limit of naked eye visibility for people with extremely good night vision is about magnitude 8.5. Neptune at opposition is much brighter at magnitude 7.7, so it should be visible to people with good eyesight. Astronomer Brian Skiff, made an attempt about a decade ago, he failed to spot it. He said that it should be relatively easy to spot from the Southern Hemisphere as it is high in the sky viewed from there.
But surely many other people with good eyesight who are capable of seeing faint stars of magnitude 8 or even dimmer who work at the Paranal observatory will have given this a try?
Well, having seen Neptune and identifying Neptune are two totally different things.
Let's tackle this one step at a time. Your link says:
From these observations it would appear that, if seen upon a perfectly black background, a star of magnitude approximately $8.5$ would be at the limit of unusually good vision.
The problem with this is that there are stars, diffused sunlight and airglow that prevent the night sky from being near-perfectly black. This has always been a problem, and it's only gotten worse with air pollution — meaning that we would be less likely to see Neptune unaided nowadays.
My guess is that in the past, there have been a few people who have seen Neptune, but none that would take note of it. Do you know how many objects of magnitude less than $7.8$ are there in the sky? The answer is, approximately 30 thousand! Good luck finding Neptune.
Even with telescopes, Neptune can be hard to point out. Galileo actually discovered Neptune 200 years before it was recognized as a planet, but he mistook it for a star. Point is, even aided eyes often cannot identify Neptune.
Anyway, the reason Curtis could see such dim objects is because he had exceptionally good vision and was aided by tools:
Without using the diaphragms and apertures, Curtis was able to see stars of magnitude $6.5$ directly against the background of the sky. With their aid he easily observed stars in the magnitude range $7.2$ to $7.4$. Four stars between magnitudes $7.9$ and $8.1$ were usually seen with little difficulty...
Without these tools, he probably wouldn't have seen it at all. Not to mention, Neptune's brightness fluctuates — at its brightest, it is $7.8$; at its dimmest, it is $8$. That might not seem like much, but when finding Neptune, it could make a big difference.
Regardless, even in ideal conditions, it is generally agreed that Neptune is the only planet that cannot be seen with the naked eye. Most sources state that you need to at least use binoculars to see Neptune.
@iMerchant That would help, but not much. You would still be extremely unlikely to find Neptune.
Well, not so bad. A quick calculation: $\sqrt(12.76/30000)*180/3.1415 \approx 1.2$, which is around 2.5 Moon diameter. It means, in the order of 2.5 Moon diameter is the mean visible angular difference between the $>7.8$ magnitude objects. And, a simple Google night sky app could show you, where you should look for it.
Here's an old (1980) Scientific American article on Galileo's visual observation of Neptune. http://www.scientificamerican.com/article/galileos-sighting-of-neptune/