These are challenges I've done:
Contests challenge: A pokemon's moves must be all the same contest type: smart, beauty, etc. For my treecko I stuck with smart. worked out for me
Moveset challenge: As your pokemon learns new moves, the newest move will replace the oldest move, no matter what moves they are. TM's are not allowed. HM's, when taught, will still replace the oldest move when taught. This is honestly the hardest challenge I've done.
Split-game Challenge: Using any games that come in threes(Ruby, Sapphire, Emerald; Diamond, Pearl, Platinum; and so on) play all 3 at once. However, once a pokemon is caught in one game, it cannot be used in the others. For example: you catch a poochyena in Emerald. You can't use a poochyena or it's evolution in Sapphire or Ruby. The games should be kept at about the same place in the story at once. It took me forever to beat this challenge.
Restriction Challenge: Starters are excluded from the following: Roll a dice and see what it lands on. Anything from 2-6. Let's say it lands on 3. Then, every third pokemon in the pokedex, including it's evolution, if it evolves, can't be used in game. The pre-evolutions are allowed, but not that pokemon, or what it evolves into. Use the region dex, or the national dex, depends on how challenging you want it to be. I used region on Diamond, and I rolled a two. I was tempted to roll again.
Random Favorite Challenge: (This challenge will most likely involve Action Replay usage) Make a list of your top 6 pokemon in every type. Make sure you have them arranged from favorite to least favorite. Then, get someone to pick 6 random types. Let's say they pick: Dark, Grass, Fire, Electric, Ground, Bug. Then, you have to arrange a team depending on what they chose. Since dark was first, use you favorite dark type. Then, since it was second, your second favorite grass, and so on. When I did this, I ended up with 5/6 being weak to electric. Though, it can be pretty fun. This is the one I do every time a new generation is made.
I made all of these myself. If there are ones similiar like them out there, I never saw them.