Hi,
The Save on Close setting should save any chat at the point the app closes or the Mac goes to Sleep. (The app is closed when you Shut Down)
They are stored in ~/Library/Containers/com.apple.iChat/Data/Library/Messages/Archive in dated folders
They are stored on a daily basis if you Quit the app or Shut down each day.
If the Mac goes to Sleep then the Messages should also be stored in the same dated folders.
This means that a long chat that is left open over several days (irrespective if the app or computer is Shut Down) are stored in several dated folders.
After that it can depend on the way you clear conversations and chats from the Main Chat window.
Mousing over the Side Bar (of Names) and clicking the X that appears is the safest.
Do NOT use the File Menu and the Delete option (or the Keystrokes listed) as this does what is says on the tin.
iMessages are also stored in the iMessages history databases in ~/Library/Messages and the three chat.db items there.
This is in addition to the Save on Close option (Provided you don't use delete the "History" of an iMessages should always come back when you re-connect to the same person).
The Normal state of the ~/Library is that is hidden.
You can copy the Paths above and then enter then in the dialogue box in the Finder > Go Menu > Go to Folder.
Alternatively you can hold the ALT key down whilst using the Go Menu and select the Library that appears when you do this and then navigate to the folder concerned after that.
The Home Folder (the one in Users with your name on it) is denoted by the ~/ as a short cut in writing the path down.
In there you should have Documents.
Music,
Movies.
Sites.
Pictures at least.
The Library is hidden as mentioned.
The Files and Folders contained in your Users Folder (Home Folder) are supposed to be Owned by you and you should have Read and Write permissions.
Do this:-
Open a Finder window.
Use the View Menu and select "Show Path Bar"
This add the location in icons and words across the bottom of the window.
Now navigate the one of the last folders in each path.
Right Click an item in the contents and choose the Get Info option.
In the Info Panel Scroll to the Bottom.
Select your name.
Unlock the Padlock.
Now check the cogwheel type icon under the list.
You should see it say make your the Owner but it should be Greyed out if you Own it.
It should say you have Read And Write permissions in the list.
You now need to repeat that for all the containing folders by Right Clicking on the icon in the Path Bar and choosing Get Info again.
9:15 p.m. Wednesday; October 14, 2015
iMac 2.5Ghz i5 2011 (Mavericks 10.9)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad