Absolute Musts:
1) Advanced Task Killer (Don't forget the widget)
2) Handcent SMS
3) Traffic by Inrix
4) Documents to Go
5) DailyStrip
6) AK NotePad
7) Google Voice
8) Google Translate
9) Google Gesture Search
10) Ringdroid
I used to rely heavily on weather and news apps and widgets, but they all became obsolete after the 2.1 update. Don't get me wrong, its not that the native widget is so great, but it is native, and it works flawlessly, and I found myself never opening any of my other news and weather apps any more.
I purposefully left out key apps that came stock with the Droid like Google Maps, Facebook, Email, Market and Browser, all of which I use constantly.
A separate mention goes to HelixLauncher which really isn't an app.
And as far as games go, I've downloaded tons, but I consistently play only Chess and Air Control