I'm often surprised at how many people load their smartphones up with time wasters like games. There were a few problem solving desktop games which appealed to me years ago but first person shooters and the mindlessly repetitive games I've seen on phones don't hold any interest for me. Instead I've got my iPhone loaded with apps which increase my productivity and I thought others might find an idea or two in this list which might help them as well.
Cloud Storage (DropBox) - My storage needs are modest so the free account suffices for me. This is the easiest method for sharing files between multiple desktop/laptop computers and tablets/phones. I don't keep any sensitive data on here long term. For times when I need to temporarily transfer such data, I create a TrueCrypt volume to store such data and transfer that up to DropBox.
Database (Bento) - Sadly, this handy little flat-file database program has been discontinued. I use it to store all sorts of information such as books I want to read and/or purchase, books that I own (to prevent duplicate purchases in paper form), house maintenance tasks, car maintenance done, etc. Since Microsoft has made Excel freely available, people who want similar capabilities might choose to use it instead. It has most of the capabilities of a simple database like sorting by different fields. If you combine it with Dropbox, you can even access your data from either your phone or a desktop PC.
Text Editor (DropText) - This gives me the ability to create and edit text files which can be shared with a desktop or laptop PC when combined with DropBox. It allows me to maintain a paperless existence at work. Staying paperless allows easier searching of notes. Making notes isn't useful if you can't easily find specific data within them.
Password Manager (1Password) - Between work and home, I have around 600 unique username/password combinations to keep track of. 1Password allows me to do this gracefully while storing them securely. I choose the WiFi Sync option so my password data isn't accessible on the cloud because I don't trust any company's cloud security.
Web Content read later app (Instapaper) - Occasionally I'll find something in a browser or one of my smartphone apps which I'd like to read later when I'll have less distractions. Instapaper lets me store these URLs to read later. It's available as a bookmarklet which works in all browsers. It also works in some of my smartphone apps.
Library e-book/e-audiobook borrowing (Overdrive) - I use it mostly for borrowing e-audiobooks from our county's library system. It keeps my 40-70 minute commute (each way) from getting boring. Buying this many audibooks would get quite expensive in a hurry.
Podcast app (Overcast) - This makes keeping up with podcasts painless. I've not found another podcast app which works nearly so well.
e-book reader (Kindle app) - Often I find myself unexpectedly stuck waiting somewhere like a dealer service department or a doctor's office where my choices of entertainment are old People magazines or whatever inane channel they have the TV tuned to. Instead I opt to read whatever Kindle book is currently occupying me at home. Since the Kindle app automatically syncs between devices, I don't even have to remember which page number was the last one I read.
PDF/ePub document reader (iBooks) - For the last 5 years or so, I've been buying all my technical books from O'Reilly in ePub format. That allows me to have them with me on my phone in case I need to look something up quickly. Searching is always better than trying to use the index in a paper book. Plus I have access to my tech books both at work and at home without having to lug them home in my backpack each night.
RSS reader (Feedly app) - I keep up with tech blogs via RSS. It lets me know when new articles are published and a glance at the title lets me know whether I'm interested enough to spend the time reading them.
Dynamic e-magazine (Zite or Flipboard) - Reading a fixed set of blogs sometimes lets interesting items sneak past me. Zite allows me to tailor the type of information I'm interested in reading. There's just enough serendipity in their matching that I'm usually amused or entertained by a few articles it finds for me.
So that's my list of the smartphone apps I use most often. Are there any I've missed which you like?