Here is a list of what you can do after the end of the game (post game content) in Pokemon X and Y. Thanks to Merosi for the list.
- The battle subway makes it's return as the battle maison. this has the same kind of challenges (single battle, super single, double, super double,...) as pokemon black and white did.
- Friend safari: Many rare pokemon with hidden abilities and great IVs for breeding can be found here. get online, socialize and start capturing and filling out that national pokedex.
- Battle Chateau: You can get a mini-rematch of the elite 4 and the gym leaders here. they'll use their signature type and the gym leaders will be higher level than they were when encountered in their gym. using writs, you can up the level further. besides these gymleaders and e4, this is also an endless supply of pokemon trainers of different levels. be sure to activate both the silver writ of challenge and the gold writ. these writs STACK.
- Online battling: With the many, many improvements to the breeding system, getting a pokemon with good IVs and the ability you want is easier than ever. going online and fighting people is easier than ever. disconnect = loss, so less people disconnecting when they feel an impending loss, making ranked battles that much more enjoyable.
- Mega Stones: Every day between 8pm and 9pm, you can go on a hunt for mega stones.
- TMs: Many of these are not obtainable before postgame
- Berry-Breeding: Your berries can now crosspolinate, so you can get mutations. collect all the berries. spread the rare berry love through wondertrade.
- Battle Institute: Get your team rated by completing a succession of trainers with varied teams. it shows your team's strengths and weaknesses pretty well.
- Looker quest: I haven't done this yet, but this seems to be a scavenger hunt in lumiose city.
- Shiny Hunting: If you're into gathering shinies, there are many ways to chain up pokemon now and increase their chances of being shiny. both pokeradar and fishchaining are in this generation.
- Daily Events: http://www.serebii.net/xy/dailyevent.shtml
Know of any other things you can do post game in Pokemon X and Y? Please post it below in the comments section. Thanks for your contributions, and for visiting the site.