I'm a pretty big fan of puzzles actually. I like games that require some thinking and skill to solve, and puzzles are like the greatest thing for that. That said, I don't really like games entirely built upon puzzles. Like those are fine, but I my personally preference are for games that have puzzles elements in them.
I actually bought a Sudoku game for my DS. I enjoy it, but I can't play it for hours on end. (Wikipedia)
I like games that have some variety in them in terms of gameplay. Random mini games here, random puzzles there, and so on. They make the game more interesting in my opinion, and can provide some different challenges to the player so that the original gameplay doesn't get stale.
In terms of puzzles, the first one I think about when I think about "puzzles in games" is the paradox puzzles in Final Fantasy XIII-2.
I've hit that Retry button so many times. (NZGamer)
I actually hated that puzzle.
It's one of the few types of puzzles they had in the game. The other ones I was totally fine with. In fact, I was ok with the above puzzle type too (It's called Hands of Time btw). However they get frustrating. Especially when in order to solve a paradox, you need to beat many rooms in a row. And there are levels where the entire level is solving many paradoxes.
I'm looking at you Oerba. (Final Fantasy Wikia)
It's frustrating when you have like 10 paradoxes to solve. And like half of them require you to beat like 5 rooms of Hands of Time. I did that all in like one sitting, and by the end I felt like my brain was about to explode.
On the other hand, I really enjoyed the other two types of paradox puzzles. My favourite of the three were the Tile Trial puzzles. In those puzzles, the player has a starting point and an end point. There are a bunch of tiles, and they disappear when you step on them. The goal is to get to the end while collecting all the crystals.
I like this one. (Final Fantasy Wikia)
This one I actually find kind of fun. It still requires thinking, planning, and sometimes even timing. Yet it's quick and not overly difficult. I didn't mind doing a bunch of these in a row since it didn't feel like I was smashing my head against a wall.
Also, I know a lot of my blog posts are on FFXIII-2, that's pretty much just because in the entirety of 2012 I played like. Four games total: Harvest Moon 3DS, FFXIII-2, StarCraft II and Kingdom Hearts 3DS. I just don't have enough recent material to work with, so I keep using the same games.