Are there any special challenges with non-trophy incentives?

I recall in Dark Souls 2 there being special rings granted to the player for beating a play through without dying and also beating the game without using a bonfire. The map of flames in Majula would also turn blue if an area were beaten eight times. Are there any incentives like these in Dark Souls 3 to do challenging feats? I'm especially curious due to the Estus Flask's item description saying "no journey of import has been made without an Estus Flask."

There doesn't seem to be any for now. DLCs might change that but there's no information out about that as far as I know. I'll update this answer if it changes.

• Getting all the rings, spells, weapons, ect require at least 3 playthroughs, but there's no other special requirements. Assuming Steam achievements are correct, that indicates there's no hidden/secret items.
• There's no way out of the various secret areas beside bonfires or using an item (which counts as using the bonfire you teleport to as far as I can tell).
• People have done no death runs and haven't gotten anything.
• You cannot respawn bosses.
• The game files have been datamined and no such thing have been found.
• Dark Souls item descriptions have always been vague lore dumps; the previous flasks descriptions says similar stuff.

The Ashes of Ariandel DLC is now out. There is still no special challenges as far as I know, but there are two noteworthy secrets.

• If you return to the friendly Corvian Settler NPC after clearing the DLC, he will give you a Titanite Slab.

• If you followed the "Lord of Hollows" questline and speak to Yuria with Sister Friede's soul in your inventory, she will have something new to say.

The Ringed City DLC has been out for a while now and still no special challenges have been found. Noteworthy stuff:

• After killing the last boss, you can return to the painter girl in the Ariandel Chapel for some new dialog.

• After killing the last boss, you can take a right from the boss door to find the ruins of a church. Shira will spawn and attack you. Defeating her will give you her weapon, and her armor can be found in the now-open room where she would talk to you.

• In the last boss area, follow the left wall to find a lone wandering ringed knight with dual greatswords. He drops a titanite slab the first time you kill him.