I've (attempted) to make a potato or sweet potato/barley hybrid beer three times.
I feel that the first two were successful, while the third was not.
The first two batches used chopped/boiled potatoes, in the same manner that one would use to make mashed potatoes at dinner time, one with sweet potato and the other with russets. The mashes each consisted of roughly 2/3 barley to the 1/3 pre-cooked weight of potatoes. I used the whole boiled potato pot to mash the barley with as well. I did BIAB, and did not have to worry about stuck mashes in this case. I used a small amount of crystal 60L in each batch as well.
What I thought was interesting is that according to my mash, my pre-boil gravities in both potato/sweet potato worts were within .002 points of where they would have been with an expected 75% efficiency. This is the thing about it- I couldn't easily really calculate my mash efficiency since how do I know was potato sugars versus more efficiently extracted barley sugars? Either way, it was horribly low efficiency, or it was typical efficiency and the potatoes added very little sugars.
I augmented both worts with DME to get to my target post-boil gravities. The post-boil appearance of both worts was horrendous in my 1G test jugs. I don't recall which yeast I used, but I believe it was a Safale or Danstar dry ale yeast pack that I split between the two 1G jugs.
The post-fermentation appearance was again horrid. I have never seen such incredible scum on the top and bottom of the fermented beer. The stuff inbetween the unprecedented trub layers was nice and clear though. I carefully siphoned the clear beer out of each jug and bottle conditioned.
As it turns out, I lost track of which one was sweet potato and which was russet. Oops. A buddy of mine tried one of each of 3+ week carbonated batches that were in bottles. He thought it was one of the best beers I had made up to that point.
I believe the fact that I used relatively fresh cascade pellets and in healthy amounts for bittering, flavoring, and aroma helped produce an attractive smelling/tasting beer. The finished beer had a nice cascade aroma, but was not overpowering. One of the two had a color that was more orange, as expected when using sweet potatoes, at least we deduced as much.
Personally, I could not detect potato essence in either beer's aroma. I felt that there was something different in the flavor, which I attributed to the potatoes. It wasn't unpleasant, but I don't think I would do it again in 10G batches to drink all the time though since the potatoes added an extra layer of complexity to an otherwise just fine simple pale ale. It wasn't worse than a standard pale, just that the extra work in using potatoes made it more of a novelty than a practical beer I would likely brew often and again.
The third batch I did used washed yeast in a starter, and whatever noble hops I had leftover after the winter. We made two batches using only sweet potatoes this time. We cut them into long 1/2" strips and baked them at 425 until they were making us salivate with visible carmelization on the tuber slices. We physically mashed them a little and used some wheat, carared, and crystal 20 along with standard 2row. Again, like in the first round of experiments, we saw much lower pre-boil gravities than expected. Despite the visible carmelized sugars on the cooked potato slices (that smelled and tasted delicious!), we either had super-low-efficiency in our barley mash, or very little sugar added by the tubers. We did between 1/3-1/2 of the pre-cooked mash in sweet potatoes.
I ended up boiling my side of the split experiment down ... down .. down. I came out with 1.060 or so. It is one of the most bizarre beers I have ever tasted. It has an unpleasant aroma and flavor that I've never experienced. I am unsure if it was the hops used (I took notes but cannot find them right now- the hops were one of the European noble hop varieties), or the yeast that was year+ old washed yeast built up in a starter days before, or the potatoes themselves in combination with the wheat/barley mixture.
Your mileage my vary, but one thing I've learned after all of this is under any circumstances, use a very solid beer as your base- not a hop-bomb or high gravity killer, just a good and proven pale recipe that is well rounded and basic with nothing flashy or overpowering. You want to be able to let whatever potato is in there to come out.