DnD has two types of rests, long and short. Long rests replenish essentially everything, and they take about 8 hours of doing almost nothing. Short rests replenish some features, but not others, and don't allow you to heal to full. Without resting at all, you'll be limited to casting literally a handful of spells, or using almost any feature, a small number of times PER LEVEL. This might be sustainable at low levels, but making it to 5th, for example, would be nearly impossible. Thus, I've decided to start with a smaller task for now.
This is much more manageable. Most classes have resources that replenish on a short rest, and you can regain hitpoints in a limited way. It's time to start planning which classes rely least on long rests. I'll only be using free material, since I'm a cheapskate.
Barbarians: Most of their abilities don't need a rest at all! Maybe we can make it with this one. Unfortunately, the most important resource for a barbarian, rages, need a long rest to get back. That's a big problem, but perhaps not insurmountable.
Bard: This class shows of our first big problem. Casters in DnD are extremely powerful and versatile, they can fill almost any role and solve many problems, and they have limited spell slots which recharge on a long rest. That means limited spells per level, which is definitely not sustainable. If we want casters, we'll have to find a way around this. Bards, like barbarians, have a key class resource, Bardic Inspiration. This one also needs a long rest. At 2nd level, bards become much more useful, since Song of Rest vastly improves hit point regen on short rests. Also, at 5th level, inspiration recharges on a short rest too. This means bards become far more useful later, but early they're a liability. I might come back to this as an option, but the limit on spells is just so damaging.
Cleric: Another caster, so limited spell slots. At 2nd level, Channel Divinity recharges on a short rest, so this is potentially viable. At 10th level, you can use Divine Intervention, which recharges after 7 days if it succeeds, but if it fails, you get it back on a long rest. For us, that means we can use it until it fails, and then not until the next level. Honestly, this class is probably too risky. This means we'll need a different form of healing.
Druid: Ah, the most op class in DnD 5e. Caster again, so limited spells. Wild shape, though, refreshes on short rest, meaning druids are probably still viable despite being casters. Also, Circle of the Land druids get Natural Recovery. This allows them to get a few spells back on a short rest. Unfortunately, this feature is 1/long rest, so while it increases our number of slots, it's not as big as we might hope. Still, druid is viable on its merits as an op class, even if this challenge cripples it somewhat.
Fighter: Finally, another martial class. We don't get spells here, so we don't have to worry about them not recharging. More importantly, all their limited features reset on a short rest until Indomitable at 9th. Since I don't expect to get that far anyway, this will almost certainly be in the party.
Monk: Yet again not a caster. Ki points, the class resource for rmonks, resets on a short rest. Way of the Open Hand, the basic monk subclass, relies on long rests, but the new UA subcalss doesn't. We're likely to see this.
Edit: finishing the rest of the classes
Paladin: Oh heck no. (Almost) everything recharges from a long rest, so it's basically a worse fighter for most of the game.
Ranger: Widely regarded as the worst class. Spells refresh on long rest as usual, so it's even weaker, but nothing else actually relies on rests at all. Outside chance that I might use this.
Rogue: Oh yes please. Only the kevel 20 capstone ability relies on a rest, and a short rest will suffice. Definitely using this.
Sorcerer: Another caster, so spells on long rest. Sorcery points reset on long rest too. This looks like a bad option, but there's one possible build here. Interestingly, a build known as a coffelock actually relies on a sorcerer not taking any long rests. We may revisit this later.
Warlock: Yet again, a caster, so spell slots recharge on... wait a second. Warlocks get spells back on short rests too! We've finally got a viable caster! Many of the Eldritch Invocations rely on long rests, but there are plenty of options there. Mystic Arcana also recharge on long rests, but we ideally won't need those often, given the level I expect to be playing at. Fiend Warlocks get sweet abilities on short rest too. This, with the fighter, looks to be the backbone of the party.
Wizard: Another caster with spells on long rest. Arcane Recovery, though, means that, like druids, wizards can recover some spells on a short rest. Unlike druids, however, wizards are limited to doing this once per DAY, not per long rest. This means, in a limited form, that a wizard can also be a reliable source of spells.
Final decision time. What do we want?
Obviously Fighter, Rogue, and Warlock. Warlock could potentially be converted to coffeelock down the line, but I wouldn't count on it. The other slot is a bit harder to fill. Another caster would be nice, but the spell limit is worrying. I'm wavering between Monk (for its short rest viability), Wizard (for the same), and Bard (for the healing/support potential).
Final verdict: I'll start building the party, but I'll leave it at three. If I get feedback from y'all on how to fill out the party, I'll take it into account.
Onwards to sleepless victory!