Maybe sticky this for easy to find?
Obviously most people have the usual things like Stanza, Spotify, shazam, etc.
A few from me.... there are obviously alternatives but :-
Wunderradio
http://www.wunderradio.com/nightstand / night stand - 2 different apps that do the same thing... a flick clock alarm clock - the former also displays weather details.
iclaimit - to track expenses - you can take pics of your receipts, record mileage, parking, etc.
classics - very much the same stanza (but quicker imo) dozens of free books
teamviewer -
http://www.teamviewer.com/en/index.aspxmobile mouse -
http://mobilemouse.com/speedball 2 - yes the game... now on iphone
numberstore - free 0800 numbers
viber -- free calls to other viber users
mywi - make your iphone a wifi hotspot, - for jailbroken iphones only i think
3g unrestrictor - lets you use them apps that only work with wifi (I use it my hava/slingbox player - as they both say they wont work on 3g - but now they do!!!)
vtie - learn how to tie a tie :oD
Redlaser - scan an item & off it goes to find if its cheaper elsewhere.
mobimileage - press start, start driving, press stop at destination & you have your mileage - handy for anyone who claims mileage expenses.
camscanner - I use this a lot to photograph petty cash receipts - it converts them to PDF & you can email them as an attachement or upload to dropbox, etc.
WorldLens - not paid for this yet - but its really interesting idea....
http://itunes.apple.com/us/app/word-lens/id383463868?mt=8epicurious - great food/recipe app
HA Traffic - live traffic reports
RAC Traffice - like above
Tubemap - London underground map / start/destination to tell you where to change, etc.
Wordsfree - free scrabble that can be played over the www & facebook I believe
Instant Heart Rate - put your finger over the camera lens & hold it to the light (if you got LED backlight on an android all the better) & it will work out your pulse.
Supposed to be quite accurate.