Really hard to narrow down to 10, but these are mine (all available from the Market & only Root Explorer has to be paid for (plus Titanium Pro, but you can use the free version)).
1. K-9 Mail or Maildroid – third party email client, far better than stock.
2. Dropbox – for sending files to/from the PC.
3. Remote PC – operate your mouse/keyboard/PowerPoint/media player via wifi/Bluetooth.
4. run Football – football (soccer) management game, highly addictive.
5. HulloMail – visual voicemail.
6. Thetrainline – find train times, book tickets (UK).
7. LauncherPro – best launcher app IMHO.
8. Titanium Backup PRO – powerful backup to SD card.
9. Root Explorer – best ever file manager.
10. Clipper – saves copied text to paste wherever – superb for logging on to websites, etc.