Oh boy, there are quite a few.
Pokémon Pinball: Ruby & Sapphire - I played this a ton when I was a kid, alongside Emerald, my first game. There's not much to say, it's pinball with Pokémon, mixed with the music, sounds, and Pokémon of RS. It had two tables based on each version, and included the version-specific Pokémon in each respective board. It had really nice original music, as well as a handful of songs, fanfares, and sounds from the original RS. It's really addicting to capture and evolve more Pokémon, and rack up points while doing so. The minigames aren't my favorite part. The Kecleon and Duskull ones are stressful at times, due to the objective being using a pinball to hit a moving target. They're a bit tedious and a little stressful from the time limit. With Dusclops and Kyogre, they often end up trapping your ball and then releasing it, wasting a handful of precious seconds. Regardless, in the end, it is pretty satisfying to beat them. Overall, this game is amazing.
I sadly lost my copy in 2008ish along with my black DS Lite and Pokémon Diamond. ;-; I soon got a new DS Lite, and I got Platinum a year later, but didn't get a new Pinball. Thankfully, earlier this year, it was released on the Wii U Virtual Console, and I've been playing it a lot. It's brought back a lot of memories for me. I really hope a lot more people give it a shot. It doesn't only go great with the recent release of ORAS, but it's also a great look back into the early-mid 2000's days of Pokémon.
Pokémon Trozei and
Pokémon Battle Trozei - As JWittz once said, Trozei is one of the most underrated spinoffs ever. Released in 2006, it's a simple, yet fun and challenging puzzle game. It is, according to Bulbapedia, similar to Yoshi's Cookie. A grid of various Pokémon heads (all ranging from Gens 1-3, all the way from Bulbasaur to Deoxys) in a box appears, and you need to slide individual columns and rows to match up three or more of the same Pokémon to make them disappear. You do this enough times to clear a level. You can try and rack up several combos for tons of points, and you'll get a big bonus if you can clear the whole grid in one go. The story mode had tons of various levels with all kinds of different Pokémon. And of course, if you didn't want to play the story, there was a nice pick-up-and-play arcade mode. The character designs were unique, and the game was filled with tons of awesome techno music. A lot of the later stages were very challenging and frantic. Overall, this was a very memorable game for me.
Battle Trozei was a sequel-ish that came out last year as a download-only title for the 3DS. It adds a couple things. Firstly, you get to drag a Pokémon to anywhere you want in the Trozei Box. This makes it a bit easier to clear the board, but that isn't the goal this time. Like in Shuffle, you "battle" a Pokémon on the top screen. There's types and type matchups implemented into the game. You rack up combos for stronger attacks. The Pokémon you attack with is the one you start a combo with. If there are several enemy Pokémon, start a combo of five or more Pokémon to attack all of them at once. There's again, many different stages. Every time you defeat a Pokémon, you can choose to use it as a support Pokémon who will always appear in any stage you play. Battle Trozei really adds a whole level of strategy to the formula. You have to really think about the type advantages here. At first, I wasn't happy with how different it was from the original, but I've come to love it. It also includes challenging, frantic stages, nice music, as well as Pokémon portraits from Gen 4-6, as well as updated older ones. You can get it easily on the Nintendo 3DS eShop, I highly recommend it.
Pokémon Mystery Dungeon games - I don't think there's much to be said about these games that hasn't already been said by the community. Possibly the biggest, most-in depth spin-off series, they take the player on a journey whose concept is as awesome if not better than being a Trainer- living as a Pokémon in a world where only they exist- and talk, too. The formula has rarely changed since the original, as it would have really been unnecessary. A fun twist of Pokémon and dungeon crawlers, basically. They have amazing music, in-depth and emotional stories, funny and clever writing, loveable characters, and all of the other wonderful things you've come to expect from Pokémon. I loved the originals, I loved Explorers, and I haven't really touched Gates to Infinity. Nowadays, I've come to realize how bare it sort of was. But seeing Super Mystery Dungeon, I am as hyped as ever for it. Featuring all Pokémon up to Diancie, every single legendary having a focus in the main story, being able to choose from all gen starters (and Riolu for some reason), it's looking to be a very promising installment in the series. I already know who I'm picking- Fennekin <3 Still undecided as of yet about my partner, though.
Pokémon Ranger - Of what little I remember of these games, they were pretty fun. A nice, alternate take on living alongside Pokémon- helping them out instead of catching and battling them. The main gameplay was pretty entertaining, the story and dialogue, as well as the music were great, too. I had nothing against the later two installments, but I personally prefer how you could have different Poké Assists of different types in the original. I honestly forgot if this was true in the other games, but regardless, I oughta go back and play them again, I really enjoyed them.
Pokémon Rumble - I never played much of these games, but they were nice, simple and fun real-time action RPGs. I always liked the origami-style Pokémon models.
Pokémon Art Academy - Again, haven't played much of this, but I'm very looking forward to get into learning how to draw several Pokémon. Even if this probably isn't the best way to learn how to be a pro, I'll enjoy using it as a fun art tool and attempt to draw Pokémon. :P
As for ones I haven't played, I really wish I got myself Colosseum and XD. I may pick them up someday.
I have to agree with most people here and say the Mystery Dungeon series has to be my favourite. I've only recently started playing the newer ones but I really enjoyed myself when I played the first few. You get to be a Pokemon! Honestly, that alone does it for me xD I was a bit disappointed though, when I heard that the personality test was removed in the recent ones because I always liked how you get to be the Pokemon most like yourself xD
In Super Mystery Dungeon (out now in Japan, next month for US and early next year for Europe), the personality test is back. You can do that, or pick a certain Pokémon. Best of both worlds :P
