Ok here are some of my favourites:
Apps
Calvetica - despite not being as fast or as sleek as previous versions, I still use this over the standard calendar
Weather HD - the most beautiful weather app
Byline - syncs with Google Reader and lets me read my RSS feeds on the go, I prefer it to Reeder for various reasons
TomTom South Africa - spend the damn money, you cheapskates, it's the best GPS app on the iPhone and has replaced my standalone GPS
iBooks - by far the best eBook reader, imho. For a cheap eBook experience, grab a copy of "eBook Search" and add TUEBL as a library (tuebl.com). Boom. Free eBooks. Only download the eBook if you own the printed copy, please, I don't condone piracy.
Frommer's Travel Tools - invaluable if you travel a bit
Photogene - great photo editing tool, but not good enough to stand alone. See next entry
PhotoForge - fills the gaps that Photogene doesn't get to
Pano - great panorama app, but I'm rapidly replacing it with the next entry
Photosynth - a Microsoft app on the iPhone, but the most amazing automatic panorama creator ever. And FREE!
IM+ Pro - the only worthy IM app, makes Beejive and the rest look a little silly. Plus it connects to Skype.
MailTones - forward all your mail to a secondary address provided, kill the default mail tones, and then you can setup custom mail tones based on sender, destination, content, all sorts
Email Caddy - ever receive zipped attachments? Open them in Email Caddy from Mail.app and you can unzip them
Box.net - 50gb free space in the cloud if you use the Box.net app to create an account before the end of December. Great for shoving stuff that iCloud won't get to.
Epicurious - the only recipe app you'll ever need. Search by main ingredient, meal type, whatever. Only downside is it needs an Internet connection...although it's lean enough to work on Edge.
Mercury - a worthy replacement for Safari, although I find myself using Safari again on iOS 5 (Reading List ftw!)
Mover+ - who needs the "Send via Bluetooth" of 2007 when you have Mover+? Send files/pics/whatever to nearby devices
Opera Mini - worst UX/UI I've seen in a while, but it works when MTN drops me down to Edge
Rove Mobile Admin - great remote admin client/SSH client/etc. if you administer servers
SoundHound - awesome Music.app replacement, and a worthy companion to the next app
Shazam - recognise music that's playing. If Shazam doesn't recognise it, try SoundHound
TuneIn Radio - the only streaming radio app you'll ever need
CameraBag - I'm torn between this and ProCamera, but to be honest, most of the time I just use the standard Camera.app
iSource - a bunch of web tools, view a site HTML source etc.
HippoRemote - the best remote control I've used. Has profiles for XBMC, VLC, and more, plus a trackpad-type mouse. Works over VNC
Carpenter - handy levelling-type functions for when you need your iPhone to help you hang a picture straight
IntoNow - like Shazam but for TV shows and movies. Let it listen to what you're watching, and it'll tell you what series/season/episode or movie it is
Regator - great for reading news and random articles when there's nothing left in my RSS feeds
Star Walk - hold it above the horizon and it'll show you the stars above you and let you tap to view info. Also has pinch to zoom. The only astronomy app you'll need.
WordBook - great dictionary/anagram solver/crossword solver/thesaurus. I totally don't use this to cheat in Words With Friends.
djay - awesome little DJ/cross-fading music player. Used to be iPad only, but is now a universal binary and works WELL on the iPhone
DeviceLocator - don't ever set it to update on changing cell towers, otherwise it'll kill your battery. But set to update every 30 minutes, and let it sync with Google Latitude, and you're sorted. Doesn't have to run in the background, just launches itself when need be, updates, and then kills itself. Has an emergency mode if your phone gets stolen, which is made somewhat redundant by Find My iPhone on iOS 5. I use this purely for Latitude updates.
Games
Angry Birds, Angry Birds Seasons, Angry Birds Rio - updated often enough that I keep coming back to it, although the game play is less skill and more luck than most games
Trainyard - great puzzle game, but basically abandoned from an updates perspective
Cat Physics - awesome, free puzzle game, occasionally updated with new levels. Worth its weight in gold.
Words With Friends - Scrabble as it was meant to be. Play against randoms.
Tiny Wings - addictive, but can lose its edge after a while. Help a bird fly and beat the sunset.
Cut the Rope, Cut the Rope Experiments - lots of fun, and quite challenging in parts. Cut ropes to feed a monster in a box.
Canabalt - had this for ages, and I still fire it up. Amazing sound track. Run and jump over obstacles, like Mirror's Edge. But better. In black and white.
Galaxy on Fire 2 (non-HD or HD version) - imagine Wing Commander if it was made in 2011. AMAZING space combat sim. Highly addictive. Don't forget to get the Valkyrie add-on for the sequel to the main story!
Infinity Blade - uses the Unreal 3 engine. Graphics are simply mind-blowing. Also: if you get bored and find it linear, beat the king and then save up to buy the Infinity Blade. Then go to the basement area and use it on the rock. New challenges! Still bored? Google a bit and research Negative Bloodlines, plus research how to get to NewGame+^2 or higher after beating the Original Ancestor. This game has a LOT of hidden depth, but people get bored because it appears to be a linear rinse-and-repeat.
Epic Citadel - tech demo of the Unreal 3 engine. Amaaaazing graphics.
Urban Asphalt 6 - fun racing game, not as hard as Real Racing
Cause of Death Free - you'll spend a bucketload on the add-ons, but a great choose-your-own-adventure type game
Game of Life - one of the better iOS adaptions of classic board games
Monkey Island 2 Special Edition - well adapted for the small screen, but it is the original in spirit and content

MI1 also available.
Need for Speed Hot Pursuit - can be fun, but can also be tedious in parts, like all Need for Speed games
Plants vs. Zombies - an addictive tower defense game...I mean REALLY addictive...you've been warned!
SimCity DLX - true to the original, but can get dull on the phone
Starfront - decent RTS, reminds me a bit of Starcraft, but it IS simplistic in the truest sense
Swords and Soldiers - fun sort-of tower defensish game, worth a few plays

Graphics are cartoon-like and well drawn
TowerDefense - grown up tower defense game set on a distant planet...extremely challenging and well made
Trivial Pursuit - awesome for playing over wifi
Scribblenauts - solve levels by creating anything you want. Giant red panda? You got it. Small green tank? Done. Blue ship? There it is. Rainbow bunny? Bounce bounce.
Sentinel 3 - decent enough tower defense game for when you've completely finished...errrm...TowerDefense
Wooords - anagram game, but really engaging, with great graphics