Lots of great apps suggested, I'll try to just raise ones that haven't been noted yet. BTW, what makes these phones 'smart' is that they're pretty much a fully-capable PC that fits in your pocket.
Files - Turns your iPhone into a wireless server, allowing you to upload/download files from another computer on the local network to the Files app. Once in Files, most documents can be read/played. Also has a browser that allows you to download from web sites and a good set of security features.
iTeleport - This one's a premium price, and worth every penny if you want a quality VNC client. Control any computer with a VNC server remotely. If you want to try out a free client, there's one called Mocha VNC. If you're a Windows user you'll probably prefer Mocha RDP. If you're a command-line warrior you'll want iSSH which also has a VNC client built-in. I guess what I'm trying to say is that remote desktop software is awesome.
Netflix - This a great service, and it has a great app. Try the one month free trial.
Air Video - Don't need Netflix because you have a bunch of videos already on your computer? Stream them to your phone with Air Video. Converts just about anything into .h264 either ahead of time or on-the-fly, and streams it to your iDevice.
Elements - A text editor that syncs with your Dropbox account. Simple, but incredibly useful for notes.
Now on to the games:
Phoenix Wright - The once rare original game, just like the DS version, but only $5.
QuickHook/HookChamp - Sort of an adventure/platformer with a great swinging mechanic. Simple controls, complex gameplay.
Texting of the Bread - A parody of the legendary Typing of the Dead. Just as fun, substantially more delicious.
Jet Car Stunts - It's like Trackmania, in the stratosphere, with a bonus rocket engine. Hell yeah.
Glyder 2 - A relaxing flight sim about collecting and exploration.
Skies of Glory - A WWII flight sim with a decent campaign and good network multiplayer.