What's the crud in the pan with potatoes and oil? Sometimes I make sautéed potatoes. I dice the potatoes into little cubes about 1cm or less to a side, then cook them in a pan with about 1-2Tbsp of hot oil. The result is sort of like little french fries. In spite of the oil, the pieces have always stuck to the pan.

I hypothesized that the potatoes were absorbing the oil and that must be the culprit, so I tried it once with about double the oil. The potatoes still stuck, but this resulted in a layer of dark-brown oily crust forming in the bottom of the pan (that was a pain to remove, too).

It seems like the best option is to use little oil, and just deal with the potatoes sticking for the first few minutes (once they get cooked on the outside, they don't stick anymore). But I'm curious: what is it that is mixing with the oil and forming this crust? And is there any better way to cook the potatoes without having them stick to the pan?
The coating you are talking about is potato starch that is browning on the bottom of the pan, similar to what happens to roux when it is prepared. If you deglaze the pan using alcohol, it will come right off without any effort (water works too, though more is needed).

As for how to get the potatoes not to stick, it's important that the pan and the oil are quite hot when the potatoes are added to the pan. As described at Newton Ask a Scientist, the hot oil encourages a steam barrier to appear between the potatoes and the pan, temporarily protecting the potatoes from sticking and encouraging the formation of a crust that will permanently keep them from sticking. Depending on the amount of potato being cooked, the amount of oil added to the pan will have to be adjusted for thermal mass, since the addition of uncooked potatoes will cause the oil to cool down very quickly.