The puzzle DLC for the Laytons, and I think the books in 100 Classic Books, are all on cartridge, so if you're willing to cheat later on down the road they're not lost
forever. Still, good enough reason for me to pick them up.
Sucks to hear that Picross DS/3D won't be able to hold all their DLC on one cartridge.
Not sure if it's worth $5 to pick up another Picross 3D just for that...
Pokemon Conquest: The episode distribution ended long ago, but you can use an odd password system to unlock them now.
Pokemon Rangers: Both Shadows of Almia and Guardian Signs still have two missions each up for download. Like Layton, though, these are all on-cartridge, so you could cheat them in later.
Dragon Quest IX: Yeah, MisterE13 is right, if you connect locally with someone who's already downloaded the quests, you'll get them, too. I wouldn't risk the chances of that happening, but it is possible.
Also: GTA: Chinatown Wars. It took me a while to confirm this, but there are two missions that only be accessed by connecting to the Rockstar Social Club and playing some minigames. The missions are only going to last you 15 minutes or so compared to dearth of puzzles in Layton, though, so I wouldn't get the game NOW if that's all you want it for.
Last bit, Pokemon. There's not really much in Pokemon that could be considered DLC (not counting Mystery Gifts, but those are all over, anyway), but there are medals (achievements) in Black 2/White 2 that are fairly exclusive to online play: Do a trade with someone internationally so you can visit Union Tower, and fill up your Pal Pad. Trading "internationally" SHOULD be fairly easy down the road if you have two copies of the game--just register your location in one copy as being in a different country, and that should work. The Pal Pad one is also
possible to get after the shutdown, but since Friend Codes won't be handed out after that date, I implore you to find 31 other copies of BW/BW2 that have connected to the Internet at some point. Good luck.
Also in Pokemon: Female Pokemon with their Hidden Abilities. Most of everybody who knows what that means has moved on to X/Y at this point, but there is still some reason to collect all the HAFs/DWFs in Black/White (RNG abuse, mainly). After the shutdown, the only way to legally get these Pokemon will be through local trading with someone who got them from somewhere else. If you have no idea what I'm talking about, then just ignore this part.