Can anyone recommend some great games for the DS? I really want to get one for Christmas this year, along with Scribblenauts. 

Since you are getting a DSi, I'll start with with some DSi games.
 
DSi games
Dragon Quest Wars- I was really surprised when I picked up this game.  It is a weird cross between chess and grid board turn-based strategy.  You select a group of monsters and try to out play your opponent by either reaching the goal, defeating the opponents monster or scoring more points at the end of the round.  You can also go online and play against others.  Costs 500 Nintendo points ($5 in USD).
 
Mario vs Donkey Kong Minis March Again!- A pretty cool game.  Rather than directly controlling the character, you manipulate the environment to guide the character.  Trying to move blocks around to have mario fall in a pit, jump on a spring, activate little mechanical to reach the goal.   You can also make stages and upload, although I have yet to try this out yet.  (800 Nintendo points)
 
Birds and Beans- Fun little arcade like game for 200 Nintendo points and really addicting.  You move around this bird and try to eat beans.  If you fail to eat a bean, that bean will destroy a part of the stage.  If a bean hits you, its game over.
 
Flipnote Studio- A free application.  You can draw and doddle and make little videos that you can upload.  Here is an example of one.
  http://flipnote.hatena.com/55CE9B7031CE9F11@DSi/movie/CE9F11_09115C1A62068_011?in=user
 Also, here is one I made.
 http://flipnote.hatena.com/5374917031D0E123@DSi/movie/D0E123_091AA71137E32_001?in=user
 
 Mighty Flip Champ- Basically a puzzle game with platformer twist.  Rather than trying to jump from platform to platform, you need to fall onto platforms by switching maps. 800 Nintendo points.
 
Pictobits- Similar to Tetris, you have falling blocks and shapes.  Although you are suppose to fill in those shapes and break up the shapes.  Then collect the bits that fall and use that to shape out the falling blocks.  The blocks are then used to fill in a picture.  500 Nintendo points.
 
I would suggest picking up the Web Browser because it is free.  It is useful for doing quick stuff but nothing you would really want to browse on.
 
I own other DSiware applications. The ones I already recommended are just some really fun stuff.  Here is a list of the other DSiware apps I own.  Feel free to ask more in detail about them.
 Art Style: AQUIA
Art Style: BASE 10
Art Style: INTERSECT
Art Style: BOXLIFE
Art Style: precipice
Art Style: ZENGAGE
Bomberman Blitz
Dr. Mario Express
Animal Crossing Calculator 
 
 Ninja Gaiden: Dragon Sword- You control the entire game with your stylus and one button for guarding.  It is pretty unique in the DS library and works really well.  You hack and slash and perform combos moving the stylus across the screen.  Tap to throw shurikens.  It is pretty simple to control and pretty intuitive.  You can pick it up pretty cheap new and used.
 
Bangai-O Spirits- This game has about 160 levels, all which are difficult.  It allows you to freely fly around the stage.  You can use a varity of weapons and devices to play the stage (although you will be restricted in the number of weapons you can carry).  Some stages functions differently.   Some stages act more as racing games than just a shooter.  Some stages don't even have enemies which focuses completely on puzzle solving.
 
 
Here is a list of DS games I own.
Advance Wars: Dual Strike
Bangai-O Spirits
Metroid Prime Hunters
Elite Beat Agents
Castlevania: Portrait of Ruin
Brain Age 2
Castlevania Dawn of Sorrow
Zelda: Phantom Hourglass
Starfox Command
Children of Mana
Lock's Quest
Castlevania: Order of Ecclesia
Contra 4
Ninja Gaiden Dragon Sword
Advance Wars: Day of Ruin
 
I'll edit this later.

One game in particular that just came out and is great is Might and Magic: Clash of Heroes. I played a demo at PAX, and it may be one of the most addicting games I've ever played.  
Besides that, there's Castlevania Order of Ecclesia, The World Ends With You, New Super Mario Brothers, and MANY MANY MORE.

• Pokemon Platinum 
• Advance Wars: Days of Ruin
• New Super Mario Bros.

If only one game series then that series is Professor Layton.

Pick up all Phoenix Wright games.

" Pick up all Phoenix Wright games. "

I'd recommend New Super Mario Bros., Advance Wars: Dual Strike, Advance Wars: Days of Ruin, Elite Beat Agents, Legend of Zelda: Phantom Hourglass and Pokemon Diamond/Pearl/Platinum Version. Great games.

" @illmatic19 said:

" Pick up all Phoenix Wright games. "

AGREED. "

Mario Kart DS, Castlevania: Dawn of Sorrow.

Get the Professor Layton games first.
Get the Phoenix Wright games second.

Here are some recommendations.
 

• Mario Kart DS
• All 3 Castlevania Games
• Sonic Rush Adventure
• Animal Crossing Wild World
• Kirby Canvas Curse
• Yoshi's Island DS
• Resident Evil Deadly Silence
  
• New Super Mario Bros.
• Pokemon Platinum
• Meteos
• Kirby Super Star Ultra
  
• Mario & Luigi: Bower's Inside Story
• Dragon Quest IV
• Dragon Quest V
• The Legend of Zelda: Phantom Hourglass and Spirit Tracks
• Wario Ware Touched

Wow thanks everyone, I really can't wait to get a DSi now. Also, does anyone know if the Drawn to Life games are good? They sound interesting.

Elite Beat Agents.

Dude there's so many, and it looks like the list is pretty well covered, but here's one I think you really need to look into: Big Bang Mini. It's a cheap game, and it's quite amazing.
 
I'm also quite into Pokemon so I'm going to throw out that Pokemon Platinum is the absolute best game in the series (although SoulSilver is looking to overtake that).

R4. 
 
I mean Warioware. Yeah, Warioware. R4 is the abbreviation for "Warioware: Touched".

" My favorite time-sinks on the DS are Disgaea DS, Final Fantasy Tactics A2, Mario Kart, Animal Crossing, New Super Mario Bros, and most of the Square RPG's - Final Fantasy III and IV and Chrono Trigger in particular. "

" Wow thanks everyone, I really can't wait to get a DSi now. Also, does anyone know if the Drawn to Life games are good? They sound interesting. "