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Downcast - The definitive podcast app - $1.99
The Apple Podcasts app has a horrible UI, is buggy as hell and deserves to be Forestalled. After the novelty of the skeuomorphic reel-to-reel tape player (remember those?) wears off, you should sober up and purchase Downcast ($1.99, App Store) -- a real podcast app. Downcast features intelligent podcast discovery (instead of a counter-intuitive "store") and controls that let you auto-download podcast subscriptions according to network (i.e., Wi-Fi), location, or time. It even streams podcasts on demand freeing up precious local storage on your device. If you own an iPhone and an iPad Downcast syncs podcast subscriptions, playlists, settings and episode information via iCloud. Apple needs to acquire the developer then ask him to redesign iTunes.
[Downcast for OS X will enter private beta "soon after the new year" which means that I might new use iTunes again.]
1Password 4.0 - A required password app - $7.99
Ever since the invention of the password there have been hacks (just ask Mat Honan). It's an imperfect solution for security but the venerable password isn't going away any time soon. In order to protect your identity, financial/health data (and baby pictures) you need to a) stop using the same password on multiple websites, and b) stop storing all of your "strong" passwords in an Excel spreadsheet on your computer. 1Password ($7.99, App Store) solves this problem by storing your passwords in an encrypted database that's protected by, one password. When paired with the OS X app and synchronized over iCloud or DropBox it's an unbeatable combination. 1Password for OS X ($49.99) is what really ties it all together, giving you the ability to create and submit passwords directly from your browser toolbar. This is single piece of software that I probably recommend the most.
Chrome for iOS - Free
I'm increasingly replacing Safari with Chrome (free, App Store) on my iPhone and iPad. Chrome has a modern UI with a unified search/URL field, unlimited tabs that you can flip through like a deck of cards and synchronization of tabs, bookmarks, passwords and omnibox data across devices that actually works. Mobile Safari on the other hand, feels like a relic of days gone by. It requires an extra touch to search, limits you to eight "pages" (no tabs), and iCloud bookmark syncing that is so broken that I leave it turned off.