These are the apps that I always come back to when I have flashed a new ROM to one of my phones
Opera Mobile the best browser on Android or any other mobile platform come to that. (free)
Beyond Pod a great podcasting app if you are into that sort of thing. (free and paid)
Power Amp - Worth the money for this music player. (trial period then paid)
Go Launcher, Go SMS pro, Go Keyboard, Go weather the whole go suite of apps are very good and are free. The launcher is smooth and highly customisable.
ICS keyboard keyboards are very personal (I know some like swype and the Go keyboard has this option as well but I just cant get on with it) and this is the most accurate one I have tried, lots of skins available for it as well. (free and paid)
Kids Place if your children want to play on your phone this app sets up a locked page of apps that they can access without you worrying about them changing settings, making calls etc. (free)
Mail Droid useful if you need to access other email accounts and not just your Gmail account. I have found it better than the stock email apps. (free)
Drop Box Cloud storage that you can access on your PC and your phone (free for 2gb and paid if you need more)
A couple of apps for fun:
Osmos HD very addictive game that looks great (paid)
CocktailFlow great looking app for cocktails and mocktails tell it what you have and it will tell you what you can make (paid).
I also always install the following apps Root Explorer, System App remover and Set Xperia, but these are for rooted phones only.