There was a puzzle in Longest Journey that bugged people where you had to get a subway key. At the time, I was so used to those kinds of puzzles, I just thought it was funny.
Adventure / Point-and-Click / Narrative Games
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I don't know if I'd say it's the worst, but it must be in the top 10: The cookie baking puzzle from Still Life.
TL;DR The author of the puzzle mocked complaining players for not having basic cooking knowledge, but I think that the more you actually know about cooking, the less likely you are to find the "correct" solution.
The player character has promised to make a batch of Grandma's special Christmas cookies. However, Grandma wrote her method in code (e.g. "a cup of love", "a tablespoon of romance"), and while the PC can remember the list of ingredients, she can't remember which is which.
Now, some things are pretty clear; the recipe isn't going to call for two cups of ginger and a teaspoon of flour. And there's only one item given without a quantity, so that must be the egg (even if it goes into the mix at the completely wrong time). But there are literally no other hints in the game, and the puzzle's author confirmed (more on this later) that you're supposed to solve it by having some real-world, basic cooking knowledge.
The problem is that the author didn't seem to know quite as much about cooking as they thought they did. Also, there are places where swapping ingredients would be perfectly acceptable (it would change the flavour, but how is the player supposed to know how the finished cookies are meant to taste?)
When I played the game, I did have some basic cooking knowledge. I'd made (from scratch) cookies, cakes, soups, stews, and even done a couple of roasts. I tried to solve this puzzle on my own multiple times. And the game won't tell you you've got it wrong until you've mixed everything and put it in the oven. I'm sure this would've been annoying enough with a mouse, but I was playing with a controller on an OG XBox, moving a slow pointer around with a thumbstick. Ugh.
Anyway, after spending far too long on it, I hit the net for a walkthrough, but in the process I found an interview with the author, in which this very puzzle was discussed. The interviewer mentioned that a lot of players hated this puzzle, and asked if the author had any response. As much as smugness can be transmitted through plain text, he smugly stated that it wasn't his fault if players didn't have basic life skills like simple cooking.
I won't go into details, but the more I've learned about cooking in general, and baking cookies in particular, the less sense the method of that recipe makes.
I don't remember if I had to look it up or guessed my way through, but, yeah. That one sucked. I don't bake much now, but I did back then and it was so annoying.
The Dig - where you have to use the drone to fix the broken mirror and you have that alien panel with some colours - it took me forever the first time to figure out how that even works, then noted down the sequence for a second playthrough so I don't have to go through that again. This was a time before every walkthrough was easily accessible online.
I just played through The Dig a few weeks ago, and can confirm. Maybe the biggest problem is that after programming your sequence, you had to wait a minute or whatever to see how the drone actually performed.
Seriously, it always amazes me how some of even the biggest titles don't seem to have had proper testing to ensure a smooth user experience. Another one that would be useful would be having at least two difficulty modes, with the option to fall back to the easier level for particularly difficult puzzles.
Oh, and back to The Dig! I remembered just now a puzzle where you need to trap an alien rodent in a little cage by herding it in just the right direction. Unfortunately, this requires waiting for it to randomly appear in the correct way, and then walking around in a specific way, with many opportunities to easily screw it up and have to start all over.
The cat hair moustache puzzle in Gabriel Knight 3 is pretty infamous, for good reason.
I came here to post that exact puzzle. That one ended the game for me.