Having just moved to a new Mac Mini on Mavericks rather than migrate from my 5 year old iMac still on Snow Leopard, these are the apps I added right away. I use a few others, but if I would be fine with only the ones listed (with commentary):
1Password (with companion app on phone and work PC)
BetterZip (The Unarchiver is fine, but I can view inside and select specific files to extract)
Choosy (allows you to choose what browser to open files/links)
Degrees (shows weather in the menu bar)
Dropbox (easy file transfers to/from work)
EyeTV (Elgato TV tuner to watch tv in the corner of my monitor)
Google Chrome (modern browsing with best extension support)
Growl (not sure if it is as useful with native notifications)
iStat Menus (see fancy usage stats)
iTeleport / iTeleport Connect (remote connections)
Microsoft Office 2011 (work)
Notify (menu bar app that checks multiple Gmail accounts)
nvALT (use for editing Simplenote files in Dropbox)
Opera (still chugging on 11.64 with customizations for main browsing)
Overflow (task bar launcher thing)
Steam (games!)
TotalFinder (from a time before tabbed Finder, not sure if it adds much now)
Vienna (RSS reader)
I'm always on the lookout for new apps that can make me more productive or do a better job so I appreciate the other lists as well.