I've completed a tier-3 run with one challenge (https://lemmy.world/post/18859193) and a tier-2 run with no challenges (https://lemmy.world/post/19011489). The idea is to play without ever using or equipping any items (armor, melee or thrown weapons) above the specified tier.
Since then, I've been trying to complete a tier-1 run, without success. Since tier-1 armor is laughable, I think there are two feasible strategies: (1) avoid getting hit, and (2) be able to take a lot of damage.
In my experience, strategy (2) is not really viable. Melee heroes (duellist and warrior) cannot really sustain the damage taken in the jail. A Monk or a Berserker might conceivably win the game with tier-1 items, but the problem is getting past Tengu. Also, they can normally rely on high-tier thrown weapons.
Going back to strategy (1), I had more success with a Rogue, making it to the first level of the dwarven city without too much trouble. I never made it past level 16, though. The ghouls hit too hard! Also, when facing four ghouls, the Rogue is just not quick enough to kill them before they revive and hit again.
This left me with one option: the Sniper. There is one problem, though: the sniper needs a thrown weapon (not her bow) to activate her lethal volleys. However, the only one tier-1 thrown weapon that the sniper can get is uncoated darts, and they cannot be upgraded nor transmuted!
My conclusion is that it is impossible to win the game with only tier-1 equipment. So I relaxed the challenge to using only tier-1 armor and melee weapons, and allowing one tier-2 melee weapon. With this combination, I was able to win the game!
I had a very lucky run with good wands, and found a couple of shurikens early on. These are ideal to avoid damage, since the sniper gets effectively four shots (one shuriken and three arrows) before the enemy can reply. Nothing except a boss can survive those four shots. I also got a projecting bow early on (just like in my tier-2 run). Projecting bows are so awesome!
Still, the run was super difficult, since at times I couldn't avoid melee combat. A succubus could take me from full health to almost dead in three hits, and dying in the demon halls is all too easy even with high-tier items.
Regarding my claim that a 100% tier-1 run is impossible, I'd love to be proven wrong. I'm pretty sure someone with more skill than I can probably do it.
A few notes. First, a shuriken is a thrown weapon (sometimes called missile weapons), not a melee weapon.
Secondly, I have two runs on record where I used only only tier 1 gear, with additional stipulations. The first was a warden run where I finished with strength 6, having used no strength potions and equipped two cursed might rings, on in the caves and one in the halls. For that run, I ended with +10 armor to avoid encumbrance, but both my +3 gloves and my +5 bow were too heavy, though I used then anyways.
The second run was a completely potionless/scrolless rogue run. I ended with the same gear I started with, and didn't end up with any additional rings or artifacts either because I couldn't confirm any were uncursed. Now, I have done potionless/scrolless with three classes total, but in the other cases I either had a lucky armor drop (leather +3 for Warden), or I found a great shield (for mage; I couldn't attack with it, but melee combat wasn't a good idea anyways so that wasn't really an issue), so that rogue run was the only one of those that was all T1.
Thanks for the correction -- I've fixed it.
I'm glad to know that a T1 run is possible with at least Warden and Rogue. I can't even begin to imagine how you were able to complete those runs! Respect.
It's also good to hear about other self-imposed challenges people have tried. Going potionless and/or scrolless is something I'll try for sure in the future!