Some of these are free, some are not (I know you said "free", but you shouldn't be put off paid apps - some are cheap and definitely worth it). You should also regularly check the App Store, as some apps get reduced or go completely free every now & then (e.g. iD Software's "Rage HD" was free a couple of weeks ago). I can't think of any decent free games myself - I have some which were free, but only as they were on offer.
Stream live Freeview TV to your iPad using TVCatchup.com (so long as you have a TV Licence of course):
Sign up for free at
TV Catchup, and then simply visit
http://ipad.tvcatchup.com/ using your iPad's browser.
BBC iPlayer official app (better than using the browser) [free]
4oD Catch Up [free]
ITV Player [free]
BBC News [free]
Engadget for iPad [free]
Skype for iPad [free]
Twitter [free]
Atomic Web Browser (alternative to Safari, has tabs & other features not found on mobile Safari, plus is better at remembering passwords) [£0.69]
Dropbox (access Dropbox cloud storage on the iPad) [free]
File Browser (access shared folders & NAS drives on the iPad) [£2.49]
iWork productivity suite (Pages, Keynote, Numbers) [£6.99 each]
Play iTunes-unfriendly videos, such as Xvids in the .avi container...
1) Sync on to the iPad with iTunes, and play with
CineXPlayer [£1.99].
or
2) Stream to the iPad using
Air Video [£1.99].
Some games...
Superbrothers: Sword & Sworcery EP [£2.99] (retro adventure game, with an absolutely gorgeous soundtrack by Jim Guthrie)
Infinity Blade [£3.99] (hack 'n' slash action, optimised for the iPad 2...)
Real Racing 2 HD [£4.99] (optimised for the iPad 2...)
Rage HD [£1.49] (optimised for the iPad 2...)
Dead Space for iPad [£1.49] (optimised for the iPad 2...)
Real Racing 2 HD on an iPad 2
Real Racing 2 HD on an iPad 2, connected via HDMI to an HDTV
Infinity Blade on an iPad 2
Dead Space HD on an iPad 2
Rage HD on the iPad 2