Unlike most conversion apps that require users to scroll through long lists of currencies to select the ones they want, this app's interface is similar to Google autocomplete. There is only a single screen, consisting of a text field and a list of results. Users type the currency and quantity they'd like to convert, and the destination currency, all into the text field, and the results update accordingly. Selecting a result displays that conversion, along with other pertinent information (e.g. possibly a map, flags, images of different countries, etc.) The app should also have a mechanism to flip the conversion to its inverse.
The app is extremely predictive and context-aware. By default, all currencies are converted to the user's native currency without even having to type it in. If the user is outside his/her home country, the first suggestion, before typing anything at all, is the local currency to the user's native currency. The app remembers and automatically prioritizes currencies the user has searched for in the past, as well as prioritizing popular currencies over less common ones. Be sure to check out the video demo attached to this proposal.
If it fits, it'd be nice to subtly show the user when the exchange rates were last updated (maybe a warning if they're outdated), and text cues when the app is doing work in the background (e.g. fetching the latest exchange rates, determining the user's local currency, etc.)