I searched through the forums and didn't see anything about this... Whenever I install CCleaner, I uncheck every box except for "Check Updates".
Since it seems every time I run CCleaner there is an update, I have to keep unchecking everything every time.
Is there a way for the app to save your installation preferences?

Updating CCleaner there is no need to uninstall the previous version.
Just overwrite it. In this way all your settings should be retained.

I don't uninstall, I just launch the new installer. Can someone tell me if the installation settings are supposed to be saved by default?
If they are, then is it possible that CCleaner is actually deleting it's own log file or some other temporary file which stores the settings?

It doesn't save installation settings like making an install.ini file with your selections to automatically select/de-select things during setup.

It's been something I've wanted for ages and requested as a feature, so if you want it too you can add in your vote here as well.