A little bit of history - I bought Diablo 3 shortly after launch, played the base game for about 2 years, and stopped in April 2014. I bought the RoS expansion and started playing again in February 2017. Since the very beginning, I was trying to get all the achievements. I remember running Fields of Misery over and over again with my level 60 wizard to fully explore all the dungeons to get the exploration achievement... Fast forward to last year (2017), I began seriously playing the game, and spend a few weeks grinding out most of the achievements. I made a Chicken Doc speed build, and made a list of all the locations that I needed to go, and the rest is history.
Here are the most memorable ones:
Historian of Tristram (read all world lore books in Act I). The hardest one is The Hanging Tree, which has a very low chance of spawning in the Weeping Hollow.
A lot of the cursed chest achievements are brutal... The most memorable ones were Cursed Cellar in Damp Cellar, and Cursed Mill in Fields of Misery.
Wheel of Misfortune (Experience all 10 possible endings of the Ancient Device event in Desolate Sands). Pretty brutal.
Nooks and Crannies (explore all dungeons in Act II). Some of those small dungeons were pretty rare, like the Sirocco Caverns.
Historian of Westmarch - The History of Pandemonium (4 parts) and The Discovery of Sanctuary (3 parts). Both of these drop in the Battlefields of Eternity. They are SOOOOO rare, and they are found in these ancient scrolls and ancient demon corpses which are SUPER hard to spot... Took me two whole days (more than 10 hours) to find all of them.
The Tourists (complete all events in Westmarch in a co-op game). Some of these events are hard to find to begin with (Westmarch Commons and Heights are pretty big maps), and I had to invite random people from general chat just to stand in and help me get these.
Haunted - Meet all the ghosts. The Dataminer and Lucious the Depraved took me a looong time.
Market Research - Meet all rare merchants. Halmin the F*CKING Alchemist, super rare merchant in Foundry Level 1, which is in turn a pretty rare dungeon with a huge map, absolutely sadistic.
All the hardcore achievements - the only reason I played hardcore. Took me a week of running Steed Charge Crusader on Torment 1.
Kill every type of uniques, rare, and champion. These are definitely some of the hardest achievements. I still remember the absolute ordeal of running through the Spider Cave for Rathlin the Widowmaker and Qurash the Reviled, The Battlefields for Mehshak the Abomination, champion Fallen Cur (which only spawns in Storm Halls), champion Plague Swarm (which only spawns in Crydor's Outpost), rare Darksky Fire Demon (which only spawns in Gardens of Hope Lvl 2), among countless others.... Also, I accidentally killed Yakara, one of the rarest uniques in the game, in Eastern Channel while doing bounties.
Kill 666 enemies with a fallen chandelier. Even on normal, monsters in a level 70 game cannot be killed with one hit by the chandelier. I had to make a low level character to do it. Took about 5-6 hours of running the cathedral over the over again.
Kill 1000 enemies using the environment. Just brutal.
Survivor (survive combat with low health 250 times). Put on Chaingmail, go to those fire pits in Halls of Agony, and grind it out. I didn't figure out a way to do it AFK, as I keep getting killed. So I had to do it by hand...
Savior (save players from death 250 times). You can actually do this solo with DH Sentry. Go to Festering Woods, set it up correctly, and farm AFK.
Bashanishu (hit Bashiok with Rakanishu's Blade). A lot of farming in the Oasis... Rip
Kill 1000 Wood Wraiths or Highland Walkers. Soooo repetitive... by the end I know the entire maps of the highlands by heart and every possible spawn points for those damn trees.
Hear Every Covetous Shen dialogue. Not hard, just stupid.
Use a Demonic Forge to kill 3 Tormented Stingers at once. So annoying.
Kill 100k demons with demon hunter. This was another really grindy one. I had a route in Act 4 which I ran over and over again. There are like 2 cursed chests in Act 4 that spawn demons, which really help.
Heal other players for 2 billion health with monk. First I was like this is just impossible. Then I started playing zmonk, and it wasn't that hard.
The set dungeons... Barbarian's Wrath of the Wastes dungeon (do not take any physical damage), and Wizard's DMO dungeon (catch 20 enemies with slow time 4 times, and reflect 200 projectiles) were the hardest in my memory.
I also looked like a fool in pub game clicking on every health pool to get that achievement...
And last but definitely not the least - COMPLETE 10,000 BOUNTIES. I just got this one today, which completes my 100%. Never again.
Getting 100% achievements was my first goal in Diablo 3. Now this is finally done (after 6 years), my next goal is to Solo GR100 with every class. So far, I've already reached 100 with Necro, Monk, and Crusader. And so it goes.
Edit: I just want to add that farming Chiltara for the Gibbering Gemstone (to make the Staff of Herding) was THE most gruesome, disgusting, and tormenting experience I've ever had. FIRST you need to get the Caverns of Frost to spawn (which happens around 40% of the time, I think), THEN you need Chiltara to spawn, which is like 50% of the time, AND THEN you need her to drop the gemstone, which is like NEVER. I farmed her for almost a full week for my first drop. Then I accidentally got a second one while doing bounties many months later. Rip.