*App prices may change, I don't plan on updating prices during sales. Some apps offer in-app purchases for unlocking more features.*
Reminders & ListsI love Clear. It's such a simple and straightforward app — so simple that Reminders were only added a few months ago, but that's the beauty of it. When it comes to productivity, I can't accomplish anything in a cluttered environment and believe me, it doesn't get any more 'clutter free' than Clear. It also syncs with Clear for Mac ($9.99).
Taking NotesI like Simplenote (free). It's free, looks great and syncs with my Mac. As the name suggests, its also simple, a common theme in this list so far. Some people prefer Evernote (free) for taking notes but I find it overwhelming, especially when all I want to do is jot down a quick note on my iPhone.
News & RSSWhen it comes to news nothing beats Flipboard (free). I don't know what it is about Flipboard but ever since I started using it, it has been virtually impossible to use another news app on my iPhone — no matter how hard I try. After many attempts I've given up and just accepted that there's no alternative.
As for RSS, I've stuck with Reeder 2 ($4.99). There are a few alternative RSS readers out there like Unread ($4.99) but I'm happy with Reeder, not much to complain about.
Every iPhone also needs a read-it-later app. I prefer Instapaper (free) but Pocket (free) is just as awesome; you really can't go wrong with either service.
MusicLocal music files are almost never stored on my iPhone, so instead I like to stream music. There are many options to choose from like Spotify (free), Rdio (free) etc. I use Spotify but all it really boils down to is preference and whether the service is available in your country. Having a Mac app is also a big deal for me.
I also absolutely love SoundCloud (free). As a big music enthusiast, SoundCloud is an incredible tool for discovering new music and following the artists you already love. I urge you to give SoundCloud a shot, I'm glad I did.
PodcastsI've been enjoying Overcast (free) lately, but there are many great apps for listening to podcasts. In fact, here are a few that Overcast recommends if you don't like the app: Instacast (free), Castro ($3.99), Downcast ($2.99), and more. I even don't mind the default Podcasts app that comes pre-installed with iOS 8. (Some will strongly disagree with that last statement.)
DocumentsI primarily use Dropbox (free) for storing documents and other important files just because I like to have those files accessible on literally every device I own. But, if you're not a big fan of cloud storage, Documents (free) is another excellent option that does a bit more and is probably the closest you'll ever get to a file system on an iPhone.
GamesI don't play much games on my iPhone, as I find other dedicated devices for gaming offer a better experience. But some of my favorites that I think every iOS device owner should at least try are: Groove Coaster Zero (free), Letterpress (free), QuizUp (free), and Threes! ($1.99). After all, you're getting the best app store for games! (And the best app store in general.)
More Cool AppsIFTTT (free) is a common "cool app" for setting up recipes like automatically downloading Facebook photos that you were tagged in. I don't currently use it but definitely give it a try, there's quite a bit of combinations that you can try out.
Another cool app, that I actually do use is TalkTo (free) which lets you send messages to stores asking anything from closing times to whether an item is in-stock or not. Very convenient and it works. From my experience, replies usually take around 5-10 minutes but I've hit the 20 minute mark before.
EDIT: The TalkTo app has been discontinued, however the TalkTo service has moved to become part of Path Talk (free).
Wolfram|Alpha ($2.99). Wolfram|Alpha completes Google (that's how good it is), so if you don't already know what Wolfram|Alpha is please try it — the website is entirely free to use. You can technically bookmark the website and have access to everything but I just love Wolfram|Alpha so much that I justified paying for an app.
If there's a category that I didn't include then I either didn't consider it essential or I currently use a default app to get the job done (examples: for calendars I use Calendar and for web-browsing I use Safari).
It's always funny to write posts like this and realize how many of my favorite apps are free, considering the sum of money I've spent on the App Store.