I don't have regular access to a computer, and the computer I sync my iPhone to has just stopped syncing contacts and events. I must find ways to back up important info online.
I depend heavily on my iPhone. I'll bold the ones that I couldn't do without. The others are awesome as well, and then I've got a lot of games that I enjoy, listed seperately at the bottom.
-Drag'n'Drop Groups
-Picture Safe
-Momo
-RssPlayer
-MailDash
-myVideos
-Facebook
-MemoryStatus
-TapExpense
-Pocket Informant
-AppSniper
-Encyclopedia
-GymGoal Lite
-Japanese
-Measures
-Wisdom English-Japanese Dictionary
-Oxford Deluxe
-Shutterfly
-Photogene
-mBox Mail
-Stanza
-Japanese Flip
-ZYB Sync
-Instapaper Pro
-geoDefense
-Deep Green
-Chess With Friends
-Wurdle
-Galcon
-Chess With Friends
-Fastlane
-Fieldrunners
-SimCity
-Super Monkey Ball
-Elemental Monsters
-Touch Touch Shapes
-Monopoly
Lots of other apps that are helpful, such as the phonics and sight words apps that I'm using to teach my son how to read. I won't list them all here though, the list is quite long already.