Maybe in an Extra Hard mode? I finished a Presentless run, and while it took me a few tries, it turns out that by abusing all the various ways of breaking aggro (pressing buttons, feeding parking meters, jumping into water, talking to friendly Earthlings, taking a hit, stepping on narrow paths and of course standing in sunflower patches and near Ghandi-ji) it can be done, and in some ways it's easier than a normal game because having an empty inventory means you have nothing to fear from Medusa Babies, Moles, Drones or Men-In-Black. It also feels like that there aren't enough entertainingly bad presents in the mix, and that $2 per identification is probably too low.
All of these things are IMO, of course, I've won at Rogue and been a multi-ascender at Nethack so you must understand I'm kind of a fanatic about these kinds of things. But if one wanted to implement a VERY hard mode, those are the things I would change. I think an endless mode would have to have some slow-but-steady means of present randomization too, because the difficulty goes down the more presents are known.