What should an endless dungeon be like?

Vaneras from the official Diablo III forums asked the fanbase what an endless dungeon should be like:

Blizzard Tracker: May 25, 2013

Endless dungeons is a topic that have been brought up quite a few times now, although I must admit I have never seen the idea combined with the suggestion of old Diablo bosses reappearing in them :-)

Endless dungeons is an idea that there are many different takes on and we have seen quite a few variations on the idea being posted on these forums and elsewhere. This idea was actually also being discussed in a recent live stream interview with Travis Day and Wyatt Cheng on Archon the Wizard's twitch stream:

Does the idea of endless dungeon come up in those conversations?
Travis: Yes, we talk about that a lot. It’s funny, with endless dungeon, it’s… it’s something people will heatedly debate what that’s supposed to mean. I have my own take, but I’m not sure what everyone wants from it. My take is that players want a measuring stick, to see how far you can go, like they reached level 200 or something. But I think the real purpose is that players just want to play for a long time without repeating the same levels over and over again. They don’t want just more Alkaizer runs.

So, to get a little more clarification on this idea, what exactly is it you have in mind when you think about an Endless Dungeon? What is it that you would like to see or find in such a dungeon? And also, what kind of gameplay and goals would you want and aim for if you could play in an endless dungeon?