Personally, while most people skip it, one I would recommend doing if you want a challenge would be the Binding Coils of Bahamut. Yes it'd outdated due to Heavensward but it's one that I, as well as a good chunk of my friends, enjoy doing time and time again... except for T9. Anyone who's cleared it can probably guess why.

The raid also reveals a nice bit of the backstory for FFXIV and from T5, T9, T12, and T13 will drop faded rolls that someone can help you turn into music... Or you can do that yourself if you level Alchemist.