DuckDuckGo lets you search anonymously. Unlike other search engines, it does not collect or share your personal information. Thus, should the feds request user information from DuckDuckGo, it'll have nothing to turn over.
I tried out the iPhone app and was surprised to find it's more than a simple search app. It's a blend of search and news. It's as if you combined Google and Newser into one app (minus the data collection and tracking, of course). By default, the app provides a search bar at the top of the screen, below which is a Newser-like scrollable feed of tiles. In settings, you can select which sources you'd like to show up in this "Stories" feed. Also, in settings, you can change it so that DuckDuckGo's Home page displays not news stories but your recent or your saved searches. Lastly, you can enable or disable autocomplete in settings.
DuckDuckGo's search function seems to work quickly and accurately, and it has a trick up its sleeve. After tapping in the search bar to enter a search term, tap the exclamation-point button just to its right. This lets you perform what DuckDuckGo calls a "!bang command," which is a funny way of describing a custom search for a specific site. DuckDuckGo lists a number of suggestions, such as !a to search Amazon and !yt to search YouTube.