I’ll take a stab at answering this.
The near-quadripoint works, you can see four countries. I’ve been there myself:
But, there are a few more candidates.
The Wakhan corridor (in Afghanistan, http://en.wikipedia.org/wiki/Wakhan_Corridor) is so narrow (8 miles at its narrowest), that I presume there will be peaks from where you can see Afghanistan, China, Tajikistan and Pakistan.
The eastern edge of Kazakhstan and the western edge of Mongolia nearly meet, separated by a 30km gap where Russia and China border. The border here is fairly mountainous, so it’s likely there will be multiple points along this corridor where you will be able to see four countries.
Technically, if you stand on the south pole, you can see land claimed by 6 or 7 countries: http://en.wikipedia.org/wiki/Territorial_claims_in_Antarctica#mediaviewer/File:Antarctic_Region.png
There’s a road that connects Igdir (in Turkey) to Nakchivan (in an Azeri exclave). The road goes through a Turkish corridor that, at times, is only some 2k wide. There, Iran is one side, Armenia on the other.
Beyond a doubt, you’d be able to see four countries here, from multiple locations.
So, the question is, besides on the south pole, where can you see five countries from the ground?