The Dialler application ?, the Messaging client ?, maybe the mp3 player? No for me at least it's :
The Contact / Address book.
You try remembering all those numbers and contact details. In my view even the oh-so-important messaging client is consumed by the all-powerful address book.
Ok, so the Dialler application is important too, but without knowing the number to dial it's useless. There's no communication without contact.
Without an address book, the modern smartphone becomes a media-player (video, mp3) or camera or just a pager to receive incoming calls. This one application is vital to all mobile phones, simply because most people will not both to commit to memory 10 phone numbers let alone 50 or more.
The thing about the address book is it your friend / family / peer / business / social / and emergency-support network all rolled into one. Any you carry it round with you.
What baffles me is why it's taking so long for the handset and OS manufacturers to offer some simple / basic functionality that's been missing for years, let alone any type of Innovation in the address / contacts book.
Multiple Sorting / Filtering / Grouping functions based on behaviour, time of day etc....are missing on most phones I see and basic algorithms to float most used contacts to the top of the list (from the call records db) is also missing.
Whilst S60 is not as bad as some, ask people how they get their most used contacts to the top of the list and they start saying things like I prefix it with an 'a' or an underscore (_).... ?!! Users should not have to make these types of workarounds. Not everyone decides to set a quickdial key even if the phone supports it.
There are some 3rd party startups like Jaiku that have started dabbling in the contact/address book space (albeit under the 'presence' moniker)- but in the meantime you can leave your comments over at Stephen Johnstone's Nokia blog about what you think could be improved ...