Hi Writer1 -
That doesn't sound like something you could adjust with settings. Instead, it sounds like an installation issue. I suggest you uninstall/reinstall Reader again to see if that corrects the issue.
I am with Donna. This sounds more like an issue with permissions to write to the registry, where this recently used list is stored (or corrupt data in that location). Sometimes normal un-install will not get rid of the things that are causing a particular problems - especially when the registry or configuration files are concerned. Because of that, Adobe has a "Cleaner" tool, which will remove any traces of Acrobat or the free Reader from you system: Download Adobe Reader and Acrobat Cleaner Tool - Adobe Labs
Run this after you uninstall Reader, and before you attempt to install it again.
I noticed that if I opened it from the Start menu that it didn't show the list, but if I opened it from the desktop icon it did. Removed the link on the Start menu, re-added and all is well. Thanks for the input.