It would be rather silly to put my own apps or wildly popular apps in the list, so I limit my choice to third party apps I like a lot and that deserve a bit of extra attention.
Easy Calendar (iPhone) - $1.99
Without a doubt the most used app on my iPhone. A calendar app on steroids. The main focus of the app is to get a task done in as little taps as possible. Solid as a rock and regular useful updates. I can't imagine ever going back to the default Calendar app.
Articles (iPhone) - $2.99
Wikipedia in your pocket with a beautifully designed interface. The developer of this app won a well-deserved Apple Design Award for Articles. Haven't tried the iPad app yet, but it's probably as awesome as the iPhone one.
Snapseed (iPhone & iPad) - Currently free but be fast
There are a gazillion photo editing apps available, but Snapseed is certainly one of the most intuitive. It features a minimalistic, touch-oriented interface. Personally I favor the iPad version. In the first place because that's where I import my photos (via the Camera Connection Kit).
Trainyard (iPhone & iPad) - $0.99
The only game in the list. I'm not into gaming at all, but this one caught my attention when I saw my children play with it. It's a highly addictive puzzle app that does a great job in the learn-as-you-go section.
Shazam (iPhone) - Free
Amazing piece of technology that recognizes any song you hear on the radio in a couple of seconds. With a useful link to the iTunes Store if you're interested in buying the song. Free with ads. Also available for iPad, but I haven't used that version yet.
Lonely Planet Guides (iPhone) - $5.99 a piece
Not really one app, but a huge collection of apps. The perfect travel companion. If you are into traveling certainly check them out.
FreezePaint (iPhone) - $0.99
Only available since a couple of weeks but I beta tested this app and had a lot of fun with it. It's hard to describe FreezePaint, you have to see it. It's a one of a kind app that's only limited by your imagination.
Remote (iPhone & iPad) - Free
This one's made by Apple, but doesn't ship with an iPhone or iPad by default. If you hate cables like me, you're going to love Remote. Browse your iTunes library from anywhere in the house. With the built-in AirPlay support the possibilities are endless if you have an Apple TV or any other AirPlay enabled device.