xplorer² remembers a lot of things about your use of the program. Most input boxes (for paths, search texts, etc) have a drop-down history, handy if you reuse the same things over and over again (you even get autocompletion, automatically or by hitting the up-arrow key). Take the addressbar for example, if you expand it you'll see all the folders you've visited recently. It also lists your bookmarked folders and frequently used folders (this is a separate list managed automatically for you, see Go to menu), and some standard folders like shell:downloads for your typing convenience — or rather to save you typing.
What if you don't want all this history baggage? You can just remove these lists from the registry, where xplorer² keeps your settings, but that's a temporary measure — the lists will be repopulated as you continue using the program. The best bet is to disable the creation of these lists using Tools > Options menu (see the pic), setting the History items maintained parameter to 2 (don't set it any lower because it will reset itself to 30).
Setting the history depth to 2 items, means there is going to be only a tiny bit remembered. The bad news is that all the drop-down lists are affected, not just the addressbar. Rename templates, keywords you search for, everything will be forgotten real quick and you'll start typing more. You must consider how much typing is your alleged privacy invasion worth.
This setting doesn't affect bookmarks and other lists under Customize menu (user commands etc). So you will not be totally castrating xplorer² <g> That's why your addressbar will still show plenty of items but they are going to be your bookmarks. If you don't want them showing up either, delete the bookmark(s).
Like the greek masters recommended, one must always seek the golden mean and avoid extremes. In private browsing may make sense in internet explorer, but you needn't worry about xplorer² too much. Let it do the hard work for you, leave the history setting to the default 30 value. You know it makes sense!
ps. A middle of the road solution is to stop saving settings on exit, from options menu.