I have no strict definition to "Atypical", but I will list some genre here for what I mean:

Clicker games
- Just click, click, click, click forever. It looks boring but they are really fun!

Dress Up Games
- It is super fun you know?!
- Love Nikki-Dress UP Queen is really great!

Hidden Object games
- They are atypical for hardcore gamers (and me), but the genre is damn popular among casual gamers today.

Super-hard and too-easy-to-die games
- Syobon Action, I Wanna Be the Guy, and many others.
- QWOP

Visual novels
- I played(or read) A LOT OF visual novels, and they are fun!

Walking simulator
- Walking is fun!

Zero-player games
- Just look the "game" runs and it is fun!
- I really loved Progress Quest.

I can list more but I would like to know more atypical games that are fun.
Maybe I will try them one day.

Cultist Simulator !

Kerbal Space Program !

Into the Breach !

FTL !

Steamworld Heist ?... It's not as weird and different as the above, but still...

Papers Please ? A tad too stressful for me.

Reigns ? Hm.. grows a bit boring, I'd say.

Ronin and Gunpoint and Mark of the Ninja. They resemble each other, but are awesome, and feel different from all other games.

Human Resource Machine ! Also TIS-100 and Shenzhen I/O

Her Story !

The Escapists !

World of Goo !

Swapper !

Incredipede ? I preferred Fantastic Contraptions, to be fair.

Also Party Hard is fun.

And Gorogoa looks cool, but haven't played it yet.

Oh, I'd also consider Creeper World as atypical. Looks like some RTS tower defense thingy, but the Creep gives it a very distinctive feel.

I love lightbot and
plan to buy Opus Magnum.
$19,99 is OK for me but I have too many games to play so I can wait. :-P

Which reminds me of (some parts of) Puzzle Agent, which I should have added to the list.

(And also Tower of Babel, on the Amiga, which I never played.)

Cookie Clicker is actually quite complicated once you get into it; it starts getting to the point where, to figure out the optimal strategy, it becomes necessary to bring out a calculator (or a program that can act as one, like python3).

Project Zomboid

Syobon Action is an interesting case; it's an example of a game in one genre (puzzle game) disguised as another (2D platformer). Basically, the way the game works is that, once you figure out where the traps are, you need to figure out how to avoid them, and sometimes (especially when it comes to the end of level flagpoles) it can be a real brainteaser.

Other games that are one genre disguised as another:
* La-Mulana (both original and remake); this game is an adventure game (and a hardcore one at that) disguised as a 2D platformer. From what I saw of a video of the Kickstarter demo, La-Mulana 2 is the same (though, judging from that demo, it seemed that the solutions to the puzzles were a bit easier to figure out without help, though you still have to think and it's still a prominent aspect of the game).
* Wizardry 4 is an adventure game (and, again, a hardcore one) disguised as an RPG, albeit an RPG that lacks experience points and only gives you levels at specific spots.
* I could argue that some of the Ultima games (particularly 4-7 at least) are really adventure games (but not as hardcore) disguised as RPGs. Of note is that these games are open world, which is something that isn't as common in the adventure game genre, but isn't something that would violate the genre definition. How about kinetic novels, which are the zero-player counterpart of visual novels? I remember watching (and enjoying) one called "My Guild Leader is a Demon". (You can find the first episode at
https://0xabad1dea.itch.io/mgliad1
Note that the video on the page isn't a trailer; it's the entire "game".)

Gorogoa is very good, but also very short.

Also, atypical game modes can be quite fun. For example:

Syobon Action's Mystery Dungeon mode (press the numeral '0' on the title screen): Procedurally generated platformer that doesn't even make a pretense of being fair. The generated levels are often unwinnable, but because the level is re-generated after each death, you can keep trying until you get and clear a winnable level (or get frustrated becaues the game keeps killing you after respawn). Fortunately, you have infinte lives in this game.

Shiren the Wanderer's Fay's Puzzles: Unlike the rest of the game, the levels are hand made and are designed like puzzles. Often, you need to use the game's mechanics in interesting ways that are never otherwise required, like using log traps to damage boost yourself across water, or even equipping a cursed item on purpose. I actually bought D.R.O.D. RPG, hoping it would be something like this, but haven't actually tried it yet.

Also there is Amok's thread : https://www.gog.com/forum/general/off_the_beaten_track/page1

(But it doesn't say which ones are fun.)