Here's my essential ones:
Free news apps like Telegraph, Guardian, New York times, always good for an idle flick through. The paid for Economist app is good if you want more depth.
Tune-in radio. You'll wonder how you lived without this. A user friendly streaming internet radio service. Plug your phone into speakers. Pause live radio etc.
Around Me. A GPS integrated app that lets you find stuff near you by searching. So if you're looking for a Tesco, bakery, pub, it will give you the closest ones and directions on google maps.
Brew Pal. Cheap brewing software and not too bad.
Spotify. Again a paid for app but you get a trial and it's a not too expensive, lets you stream hundreds of thousands of tracks. Type in an album you want to listen to, plug it into your speakers.
Games: Doodlejump, Plants vs Zombies, Monopoly
For fun but not really useful: Google app (lets you google search using a photo you have taken), Shazam (hold your phone up to music and it will tell you what the track is), Word lens (translates foreign languages in real time through your camera, has to be seen to be believed, point it at a foreign food menu and it will show you in English).