Hello there,
I stumbled upon some game breaking bugs on my first playthrough of the Legendary Edition of Skyrim for the Xbox 360. The bugs hit me all at once, which is probably a good thing.
[1] - The first issue that arose was that I noticed that Farkas and Vilkas would NOT stop engaging conversation with me, forcing me to turn to them to talk. This would stop me from combat and cause me to focus on them which had me rushing through the main quest for The Companions.
[2] - The next bug that came in was the "Elemental Fury" shout which I used quite often. For some reason I must have done something in the wrong way at the wrong time to cause this bug: The sound effects of Elemental Fury shout would ALWAYS be active, causing a loud whirlwind sound that never stopped wherever I went.
[3] - The third and most game breaking of all of the bugs I have discovered is...locked character level scaling! I was level 31 and not fighting foes that would give me a challenge for Aela and I; I played on expert for a while and since I was becoming too powerful I balanced it out by switching to master but this became irrelevant when enemies would not scale to my level.
These bugs had me conclude that the best thing to do is recycle the character that I spent around 55-60 hours of gameplay. On the upside the bugs did allow me to figure out a way to conclude my Archer Dragonborn Werewolf who went on to lead The Companions as the Harbinger. I didn't get to finish by marrying Aela and moving into the Lakeview Manor but I do feel satisfied with my FIRST playthrough of Skyrim with the Legendary Edition even though I did end it due to game breaking bugs.
*I did NOT experience Dawnguard or Dragonborn DLC's so I still have quite the adventure ahead of me!*