What kind of apps are you looking for? I have the touch, which is the iphone except slimmer and without the phone bits. I have combed through a lot of apps, so I'll list some of my personal favorites. Most of these are free apps, too, or at least have free versions that you can get.
Handy Tools & Everyday Stuff
Dictionaire - basically a dictionary that doesn't take up a lot of storage space. It's nice to have if you need to find out the meaning of a word on the spot.
WiFinder - since you have the phone, you probably aren't as worried about finding a wireless network, but if you are, this one works well in detecting wifi around you, better than the built-on one.
Beer Brands - This one is awesome, has a giant database of every beer you can think of, with details about each beer.
They also make Drinks/Drinks Free, which is similar to Beer Brands, but tells you what ingredients you need and how to make all sorts of drinks.
myLite - Works well if you need a flashlight/custom flashing emergency lights/etc. Not something you'd need all the time, but I've found times where it was useful.
Compass Free - If you happen to be lost in the forest or something, then this thing works like a sundial to determine which way you're going. Kind of useful...
Lose It! - if you're into dieting and keeping track of crap you eat, this thing has the potential to be extremely handy.
TiltMeter/TiltMeter Pro - Basically a level. It works very well, and I put it to good use while working on renovating my bathroom. Also nice for hanging pictures, etc.
iHandy Level - works similar to the above, but resembles a traditional level. Paired with TiltMeter, it has it's uses.
iChoose - just a random coin toss/dice rolling/card choosing gadget that helps you make up your mind.
Shazam - Probably one of the coolest things you can do with the iphone. It basically can listen to a song on the radio using the mic, and after a couple seconds tell you the song, band, etc. So if there is some song playing that you really like, you can use this to figure out who it is.
AroundMe - GET THIS! Can find restaurants, hotels, coffee shops, theaters, and much more wherever you're at. Tells you locations, gives you links to their websites, it's just damn awesome. If you're going to get any of these, get this one.
Google Earth - Yes, they have Google Earth for the iphone/touch. It works well, so if you like it, get it.
CoolIris - Yes, they have this as well. Haven't really used it yet on my touch, but it seems to work well.
Maps - Google Maps, which is another extremely handy tool. It can pinpoint right where you're at, and you can plot directions. It can work as a GPS unit, giving you instructions on where to go. Highly recommended.
Translator - just a little translator, if you can't understand the guy who you're ordering your food from.
BofA - if you use Bank of America, this is awesome. Get it.
GamesThere are tons of games for the iphone, here are a few that I have and like enough to keep. Not gonna write up any details on the games, but I'll categorize them according to the type of game they are. They aren't organized perfectly, but I'm not getting paid to do this, so meh.
Word Games/Puzzles:
Sudoku
WordWhirl
Blanks
Word Warp
Hangman
WordFreak
xCube/xCube Lite - Virtual Rubics Cube
Board/Board-like/Arcade games
Checkers
Touch4
Dots Free
Morocco
i Say/Free - Simon Says
Knots - Twister
mazeFinger
Dactyl
Labyrinth LE
MarbleMash
iBall3D
reMovem free
FireDrop
TiltSnake
iBricks/Lite
Space Buster
AstroTilt
BixLite
Topple
Trace
Action/Vehicle Games:
tapDefense
Paradox Lite
S.deadbeef
Aerolite/Free - basically asteroids
Reign Free
iShoot/Lite
CannonGame
SinkSub/Lite
Crash Landing
CometBuster
Cube Runner
Tunnel
iSniper/Lite
LightBike - Tron, anybody?
Tank
Scribattle/Lite
Heavy Mach/Lite
FallingBalls
Sport/Other Games:
aqua hoops, fun, if a bit simple. Based on those little handheld water-pump basketball game things
Bowl/Lite - arcade bowling
VirtualPool/Lite - Very cool, works well
iFishing/Lite - also very good
Darts
iBowl
BeerPong
FS5 Hockey
Fish Tycoon
Tap Tap - If you like dance dance revolution, or similar games, this is that, only with your fingers...
Music Tool/Games:
PocketGuitar - pretty damn awesome, considering.
DigiDrummer - fun for you, annoying as hell for everyone around you!
MiniPiano
FreePiano
Totally Random and/or Stupid Crap that is Nevertheless Amusing to Some:
Banner/Free
Rimshot!
Grenade
Cradle
BubbleWrap
Wooo Button
Zippo Lighter
iSteam
iJiggles
iNeko
Binary Clock
PBJTime
Tally Counter
Sound Box
Dog Whistler - this one is kind of fun, since it's adjustable
Planets
Okay, you'd damn well better try some of those, I just spent an hour typing all that out.
And don't stop then, because there are thousands more apps that you might find useful, and there are even more that cost money. If you don't mind paying, for every one I listed above that is free, there are dozens more better versions that cost a buck or more. So, have fun with your iphone.