Backstory: Throughout history there is a long tradition of wars starting because men jumped the queue – the Dance of Dragons is no different. Around 170 years after Aegon's Conquest, King Viserys I Targayren had a daughter – Rhaenyra – and no sons, so he trained Rhaenyra to be his heir. Great. Unfortunately, and predictably, Viserys went on to remarry, this time he did manage to produce a son. You can probably see where this is going. The Targaryens, and the Seven Kingdoms, were split down the middle as civil war nearly brought Westeros to its knees. Cool, huh?
When: Around 170 years before Game of Thrones.
Connections: Again, there would be no characters we've seen before, though the Dance of the Dragons has been referred to a number of times throughout Thrones – most notably when Shireen Baratheon tells her dad, Stannis, all about it. Just before he burns her at the stake.
Pros: In terms of drama and intrigue, there are very few stories from GRRM's universe that are quite so appealing as the Dance of Dragons. Sibling on sibling. Dragon on dragon. What's not to love?
Cons: As with Aegon's Conquest, one of the biggest strikes against this idea is purely financial. You can't make a show that is actually about dragons without, you know...including an absolute fuck-ton of flying, fire-breathing, lizard-y things at every opportunity. That, sadly, comes at a cost. That said, CGI technology improves on an almost daily basis, and given that HBO already knows that any story based in Westeros will allow them to practically print money, perhaps it will be a price worth paying.
A show about this period of Targaryen history would have everything Game of Thrones has, and more. It is without question a frontrunner, but might just involve HBO making a little trip to the Iron Bank first.