Why does gamed contact so many IPs and Ports? I've been running Little Snitch for a month or so now and it has been quite fascinating to see (and block) so many attempted connections.

gamed is constantly 'in the news'.  Please check out the attached screenshot for gamed.  I've been allowing connections as it is an Apple certified process.  Is it meant to be hitting so many IPs and different ports?  What's more, I do not play games on my MBP, though my kids play through my account on iOS devices so perhaps that is where the activity stems from.

Is gamed meant to be hitting so many different IPs and ports?
Yes it is. Why it does I couldn't tell you, but it does seem to do this as per this Apple forum. There are ways to disable it if you would like it to stop being so annoying.

From that same Apple forum:


  If you're monitoring network activity using Little Snitch, though, it's very annoying because it continually generates network traffic, whether you are using Game Center or not -- and it won't take no for an answer.
  
  You can prevent gamed from running by logging in as an administrative user, running the Terminal application, and typing (or copying and pasting) at the prompt:
  
  sudo defaults write /System/Library/LaunchAgents/com.apple.gamed Disabled -bool true
  
  After restarting your computer, gamed will not be running and the Little Snitch network monitor won't be flashing.
  
  To reverse the change, you can type:
  
  sudo defaults delete /System/Library/LaunchAgents/com.apple.gamed Disabled


Edit: also note that because little snitch manages your computer's connectivity, not your network, the gamed process that is requesting those IP addresses is the one on your computer, NOT one from any iOS device.