As always, let me point out that what works for me may not work for you ... but it may give you a starting place, or an idea, or simply to cross-off something you've been thinking about for a while.
Pages - The app I use for taking notes with my iPad; it's a robust word processor that allows for simple sharing and storage. I don't like typing lengthy pieces of work on my iPad, but when working up ideas for writing projects, it's perfect.
Kindle - I use the kindle app to ... wait for it ... host and read books. A big part of a writer's job is to read, and the Kindle app is wonderful for that; it also allows me to search through my earlier books for specific words or names or places, much more quickly than flipping pages.
Google Maps - Although I tend to have a pretty good idea of the places and spaces through which my characters move during the course of my stories, I love being able to check directions and distances and relationships between places and objects with this superb mapping app.
Sunset & Sunrise - My characters and I like knowing when the sun rises and sets in different places, and this is a wonderful resource that allows us to do just that ... I also have two that help me with the phases of the moon and the tides, but that's too nerdy to admit, so I'll pretend this is the only one I use.
Netflix and Amazon Instant Video and HuluPlus all come in handy once in a while, when I refer to a TV show or a movie in my writing, and want to make sure that I have the quote or scene or characters or action reference correct.
Pandora internet radio is great for picking/playing music while you write. I always write to music, and it's almost always instrumental classical music (no singing, and no hard edges to the music). I tend towards Mozart and Bach, but have gotten lucky quite often letting Pandora take me where it will.
Voice Recorder is my favorite of the numerous voice recording apps that I've tried. I wanted a simple interface that I could operate easily with my iPhone and sausage-fingers while driving (I get lots of ideas while on the road); it also arranges the files in a logical manner that works for me.
Camera+ is an easy to use camera app, that offers a little more control than the basic camera that comes standard with my iPad and iPhone. I like taking photos of places and people and buildings and documents, for later use in my writing, and this app makes it easy to do.
Wikinodes and WikiLinks are wonderful research tools that deliberately shows the linkages between subjects in a visual web of informational nodes. I love starting on one subject, and seeing where following the info-nodes takes me.
Date & Time Calculator is a useful tool for figuring out all manner of conversions in dates and times. I find the need for this in every writing project, and this app makes it much easier than ether googling or using pen and paper.
Pomodoro Tmer is a fantastic, and infinitely configurable timer, based on a proven productivity research for maximizing efficiency to help you beat procrastination in long-term projects like writing.
Writer Lists is a treasure trove of lists of information that can help a writer with all sorts of details for their writing: character names, physical traits, occupations, milestones, and personality; plot, setting, genres, along with a ton of other information useful to writers ... all gathered in one place.
I'm certain that there are other ways to get the stuff done that I am able to do with the help of these 13 apps, but having them loaded onto my iPad and iPhone make my life, and job as a writer, that much easier, and leave more of my brain available for thinking about writing stories.