From the bottom up:
Drafts - This app is my go-to now for virtually anything involving text on my phone. I'll have more on this app in a post this week (maybe even a screencast) and you'll see that using this app for sharing any kind of text between apps is awesome. From Dropbox to Messages to searching Google, it's become my go-to app of choice. More on this great app later this week.
In the 'Get It Done' folder
Dropbox - new UI update a few months ago makes it easier than ever to use.
Google Drive - Use for all my Google Docs, and today's update even features widescreen editing of Spreadsheets. Nice.
Cloudier - Cloudapp client for links, images, and text sharing. I use it for all my links as it's tied to my Twitter/Tweetbot.
Instashare - Wonderful too for moving files instantly from one device to another,. For example, the screenshot photo of my iPhone in this post was move by opening the app on my phone, dragging the picture out of my Library, placing it on my Macbook icon with my Mac running the same free software, and my 1.3 Mb image was transferred over Wi-Fi instantly. Great for files you would normally email to yourself.
Teambox - Great little service for teams up to five users. I use it to coordinate with Hashtag Media peoples. Project management, to-do lists, conversations - great for small teams. Paid tiers as well for bigger companies. iPhone, iPad, and web apps. Web app works great in Fluid as a dedicated app on the Mac.
Elements - My plain text repository. Where I use Drafts for quick text and lists, I use Elements for long-form text like class and teaching docs. Syncs with Dropbox and talks to Drafts as well. Anything I compose in Drafts can be directly shared with Elements.
Mailbox - Been using the hot new mail service for about two months now. Very good. They've worked out some bugs, and the service is great for eradicating and acting on every email.
Feedly - Since Google Reader is shutting down, I just moved all my feeds to Feedly. The app has great design and will seamlessly work for you after the Reader shutdown on July 1.
Sunrise - wonderful calendar app that syncs with Google Calendar. See my previous post for a review.
Instacast - There's a lot of podcatchers (podcast-catching apps) out there, but in my opinion, Instacast is the best. iPad version as well.
Tweetbot - greatest app for Twitter you'll find on iOS, both for iPad and iPhone.
Instapaper - great service and app by Marco Arment that simply lets you save articles for reading later. Safari and bookmark extensions available as well as integration into Feedly.
ScoreCenter - ESPN has greatly improved this app from it's beginnings in 2009. Now supports the iPhone 5 and will let you make favorite teams a get push alerts.
In the Photo / Video folder:
Snapseed - this is a great little Google app that lets you do lots of simple stuff with photos. Not bad for free either.
Vine - the Twitter-owned 6-second video sharing service. Great UI. When teens figure out what this is, they're going to be excited.
Redbox - Redbox redesigned their app just last week, and it's prettier, faster, and much easier to use. Redbox is everywhere now and is very convenient.
Google Maps - I never actually used Apple Maps on my phone, I've always stuck with Google Maps. Improvements have made this app great in the past year.
Forecast - This is actually a web app that runs like a App Store app. If you go to Forecast.io in Safari, you can "Install" it on your device. It's a great weather app from the makers of Dark Sky. Wonderful UI with little animations and a dedicated iPad version as well.