The Danube

As far as I can make out, the Danube only actually flows through six countries; but it forms borders to four others.

It rises in the Black Forest, in Germany, and then flows through Austria. It forms part of the border between Slovakia and Hungary, then flows southwards through Hungary to pass between Croatia and Serbia. It then turns eastwards again to flow through Serbia, and when it comes to Romania it forms the border (with Hungary). Further on, the land on its southern bank becomes Bulgaria, after which it turns northwards to pass through part of Romania. When it reaches the city of Gugliestu, which is in Moldova, it is joined by the Prut, and promptly turns its back on Moldova, veering south–east. At this point it forms the border between Romania and Ukraine. About 40 miles further on it begins to form a delta, the northernmost branch of which continues to form the border between Romania and Ukraine. Some branches then enter Ukraine, before they all flow into the Black Sea.

Of those ten countries, the four that the Danube doesn't actually pass through (as far as I can determine) are Slovakia, Croatia, Bulgaria and Moldova. Its drainage basin extends into nine more countries: Switzerland, Italy, Czech Republic, Slovenia, Bosnia & Herzegovina, Montenegro, Albania, Macedonia, and Poland. Or ten if you count Kosovo.

Wikipedia has maps showing the course of the Danube and its drainage basin.