So, I've been browsing the help forums on multiple websites over the past few days and have come to the conclusion that there is no legitimate fix for the Xbox 360 digital version of the game for the reoccurring Beagle error code. I can only imagine the root cause for such repeated errors, but the reality is that this is unique to Destiny, not my hardware, as I've played multiple downloaded games on my Xbox 360's HDD before without any issue resembling this. What does frustrate me is the apparent lack of direct developer response in the forums to this issue, stopping at the redirect to the page telling people to reinstall it (which many of us have done repeatedly to no avail, with some people going so far as to doing factory resets that still did not work).
To those of you who find a fix for Xbox 360's digital copy of this game, please do tell me, as the current "fix" in the forums only allows me to play the game for 20 minutes before the same "unreadable disc error" appears again.
I have my faith in Bungie, I love them to pieces, but this is seriously hurting my trust in them as developers who take action to assure my enjoyment (or at least opportunity to play) their game.
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For what its worth, it is also occurring for me on Xbox One. Haven't yet tried reinstalling.