The apps make the iPhone. So every time an acquaintance picks one up, just about the first thing they ask is: "What apps should I get?"
This is my current list of apps you might want to download. It's tailored to my use case, which is that of a fairly frequent business traveler and social media user. To the degree your profile is different; your mileage will vary. I've left off games--a post topic of its own--but I've included some honorary mentions that are good and innovative, but which I just don't find myself using as frequently as some of the others.
TweetDeck (free). This is my favorite twitter application for the iPhone. Tweetie 2 is another popular option.
Facebook (free). You can link Facebook into TweetDeck, but I prefer using the standalone application.
Google (free). In addition to typing in a search in the usual way, you can also try speaking your queries; the results are predictably imperfect but they're accurate enough of the time for the voice option to be very handy. For certain queries, Google can also use your current location to refine results.
Kindle (free). Reading on the iPhone works surprisingly well given its relatively small screen. That said, I view this app as primarily a companion to Kindle or iPad hardware. You might read a few pages of a book while waiting in line or taking the subway, but probably wouldn't buy a book with the intention of reading it exclusively on an iPhone.
Yelp (free). Yelp's crowd-sourced reviews of businesses have their critics, both for the uneven quality of the reviews and for the (unproven) accusations that "pay for play" benefits for the businesses that are Yelp customers. Personally, I find Yelp a useful resource and, combined with location information from a phone, an invaluable in a city or area that I don't know well. AroundMe (free) is similar to Yelp in many respects. Sometimes I prefer one and sometimes the other. Get both. Zagat To Go ($9.99) is an economical iPhone version of the popular and pithy survey-based restaurant guides.
Wikipedia (free). Wikipedia works well enough in a browser on the iPhone, but I want some nugget of information when I'm on the road often enough that it's worth it to install a dedicated app. This is the Wikipedia Foundation's official version, although there are a number of similar alternatives as well.
Fluent News Reader (free) is probably the best news application for the iPhone that combines feeds from a number of different sources. I'm also a fan of the NYTimes (free)--although it's less distinctive than the dedicated iPad version--and TIMEMobile (free).
Epicurious (free) is the recipe application for the iPhone that I find myself trying first. Perhaps this is because it includes the recipes from the late, lamented Gourmet Magazine. It's a great way way to find a recipe when you're in the store and see some nice cut of meat or fish on sale.
Pandora (free). The popular music streaming service is not only great to use on-the-go but, plugged into a home stereo, gives you an easy way to play customized Internet radio channels at home as well.
Tripit (free) is a service to which you mail your flight itineraries, hotel reservations, and so forth and, all going well, it will parse them into a standardized format. The iPhone app displays this and other information you enter in an easy-to-read format. I find it invaluable to have my travel information at my fingertips in an organized way.
i41CX ($7.99) is a faithful recreation--well, except for the great keyboard feel--of one of HP's classic calculators. If you can't conceive of paying money for a calculator app (and $24.99 for the i41CX+ version), you're probably not a prospect for this. I don't really need this, but I strongly prefer RPN and I used an HP-41CV for many years so this is just comfortable and familiar.
RedLaser ($0.99). You walk into a store and some gizmo you've been thinking about buying is on super-special-limited-time-only sale. But is it a bargain? Scan its bar code and find out what it's selling for on the Internet. I can't say I actually use this app all that much, but it wows me that such a thing is possible.
SoundHound (free and $4.99 versions). What's that song playing on the radio? Let SoundHound listen and tell you. This is another app I don't really use a lot, but makes for another jaw-dropping iPhone demo. Now that there's a free version, there's no reason not to check it out.
Evernote (free) "lets you create notes, snap photos, and record voice memos that you can then access any time from your iPhone, computer, or the Web." It's a service that has developed nicely over the past few years although, for some reason, it's not something I use systematically enough to be part of my regular work flow.
What applications do you find especially useful that I didn't mention?