The biggest issue is what defines a road, a country and where a road starts or ends.
That being said, there are two likely candidates:
Option 1: The Pan American Highway:
The Northern Pan-American Highway travels through 9 countries:
United States (interstate system official)
The Southern Pan-American Highway travels through 8 countries:
Important spurs also lead into 4 countries:
Option 2: AH1 (with the E80 extension):
This travels from Japan through Hong Kong, China, all the way to Turkey, and connects to the E80 to Portugal, connecting over 20 countries. However, connecting over the ocean is interesting, as is joining two roads, and whether or not you count countries like Hong Kong as actual countries.
However, I'm pretty confident with whatever mechanism you use to measure, the outcome will be one of the above two options.