I've done four in a day more times than I can easily remember in the normal course of business. Three in a day I probably couldn't even count.

Five in a day (with actual work to do in each of those) is a little more difficult to recall, but I've done it a few times most recently just 3 weeks ago (India - UAE - Ghana - Algeria - UK).

Also, some years ago I did Switzerland - Liechtenstein - Austria - Germany - France - Italy with some friends but this was just a touristy trip with no real purpose other than to use our rail passes to go as many places as possible. I managed to get stamps from each of those except Germany as well!