Is there a prohibition to eat the end of bread? Is it only the very first or last piece or any piece with extra crust on it?
There is no prohibition to eat the end of the bread. In fact, there are numerous poskim who say that it is permitted as there is no source for it. There are however people that have the custom not to eat the ends of the bread, and they say that it causes one to forget torah, and therefore are careful about this. Those who have this custom should stick to it, however it would only apply to a very small amount on the two sides of the loaf of bread.
As a side point according to some poskim, this minhag would not apply to eating bread on Shabbos or by a seudas mitzva.
Poskim who say to keep this minhag if you have it- Minchas Yitzchok 9-8 (7), MIshne Halachos 11-148, Tamei Haminhagim 176 however see see Piskei Teshuvos 167-3 and ftnt 29. On the other hand see Orchos Rabeinu 3 pg. 104 (old edition) 4pg. 143 (new edition), Doleh Umashkeh pg. 362, R’ Yisroel Belski zt”l, and B’nisovos Hahalacha 46 pg. 581 qouting R’ A. Auerbach shlit”a, who say that there is no need to be careful about this. Also see Kovetz Mbais Levi Inyonei Yora Deah pg. 96, and Sharei Habracha chap. 3 ftnt. 96 regarding seudas mitzva and Shabbos. Also see Aliba Dhilchoso vol. 30 pg. 58.