Just a few things I use on my phone:
Timecode calculator for quickly adding up editing runtimes- saves me a ton of time with work. I remember when a decent standalone TC calc cost buko bucks. Now it's a free app.
Tip Calc, Unit Converter, various other calcs... 'cause I suck at math.
CamScanner. Love this thing for quick scanning documents, checks, any info I need to save while on the go.
IP Cam Viewer: IP camera lets my wife and I view our son anywhere, right on our phones, and no need for the dedicated babycams.
Sketchbook Mobile- for quick sketches, jotting down ideas, layouts, etc.
Note Everything but recently switched to Flick Note; synchs my Scrivener writing projects with Simplenote. I can make edits and changes right on my phone, synch is taken care of.
S-Note I now use all the time- pops up on S-Pen removal.
Jump Desktop lets me remote-manage my home and work machines.
Shazam & SoundHound- got both either free or a 10-cent sale. Thought I'd never need these, but I'm amazed how many times either has come in uber-handy. Once had a producer call me asking, "What was that (temp) music used in this sequence, we want to get that for the final." HellifIknow- but SoundHound did, no problem. Been a bunch of times I've been out, heard a track I actually liked, and was able to easily ID it.
Aldiko Reader- not only for books, but it also the fastest PDF reader I've yet seen, so I use it for reading storyboards and scripts. Also Kindle, ComicRack, Marvel,
Dropbox- in-freakin-valuable. I pretty much save/cross-synch everything to DB- FinalCut project files, writing and editing projects, work PDF files etc. etc. I have everything redundantly backed up (including on my home server) instantly.
All the usual apps in the car: dock, BeyondPod, local/online audio players, GPS nav/info etc.
The usual entertainment apps: Netflix, YT, DicePlayer, N7Player, Dsub (for streaming from my subsonic server) etc.
Browsers, FTP, email, the usual web-stuff.
Photo/video taking apps, QuickPic, usual media apps.
Google Calendar and Tasks, synched with other schedule/task/alarm apps.
Mint.com to keep track of finances. (I love seeing *exactly* how much my wife just spent pop up right on my phone so I can send her a snarky message before she's even out of the store). Also stock apps.
Pulse, Flipboard, Currents, various news/info aggregators.
All the usual communication apps: dialer, contacts, Voicemail+, MightyText, etc. etc.
Tons of games and emulators.
Various system tools, backup, customization etc.
I'm always open to new apps that make my phone even more useful/entertaining/valuable to me.
I have to say, I don't understand how some people can go on endless eye-glazing rants on phone spec minutia... then turn around and use one like a dumbphone. What's the point?