As of the release of the Ascendancy expansion, the Perandus leagues has been the most successful challenge leagues, and it seems that the players have become especially fond of it due to the amount of Path of Exile items to be had there. The story behind the Perandus leagues is quite interesting as well, which perhaps matches Ascendancy weight and substance.
These leagues are based on the Perandus family, which is sort of like how the Medici family were in real-life 15th century Florence. They were a powerful family that controlled a vast majority of the Eternal Empire's wealth, but were then affected by the Cataclysm. Cadiro Perandus—an NPC in the leagues—was able to stash away much of their treasure in secret caches spread throughout Wraeclast, which can now be found by players in zones after defeating their guards.
Players can take advantage of Cadiro Perandus' greed for coin by exchanging Perandus Coins for items. Those Perandus Coins are why you'd want to look for those caches in the first place, and you can also trade with other players for them in the hopes of getting better items from Cadiro. He actually has quite a few PoE items that may be of good use to you.
While there's a lot more to the background story of the Perandus leagues, which is something that Grinding Gear Games tends to do to expand the in-game lore, let's get to what the leagues have to offer for now. Aside from the usual challenges and other commonalities with other leagues, there are the aforementioned items. The unique items were designed for these leagues, and they're indeed hard to get since there's nothing that affects Perandus coin acquisition.
They were designed to not just be nominally powerful, but great enough to be worthy of myth and legend. For instance, there's the Varunastra, a Vaal blade inspired from Hindu mythology. It's a one-handed sword that has very good range of bonuses and a potentially strong one in making the weapon count as all one-handed melee weapon types. That last part lets you pile on the passives and other bonuses for different one-handed weapon types.
The Trypanon is a two-handed great mallet that was inspired by the obsolete medical practice of trepanning (boring holes on the skull). This thing makes all your attacks count as critical strikes, although it does have reduced attack speed. But if you build a character right around it, then that shouldn't be a problem, especially with the increased stun duration.
Zerphi's Last Breath is a curious-looking grand mana flask that was inspired by a minor character in the Sacrifice of the Vaal storyline. As Zerphi had maintained a youthful appearance by slaughtering innocent Vaal citizens, this flask lets you gain an amount of life equal to the mana cost of a skill used multiplied by a big percentage (800% or so) while this flask is in effect. This is an incredibly powerful item that can be a great asset in Hardcore play when mastered.
There are other two unique items, namely Seven-League Step that simply grants increased movement speed and Umbilicus Immortalis that lets flasks apply to your zombies and spectres at the cost of not applying to you. These are pretty much why players are going nuts over Perandus, and for good reason indeed.