Taiwan

Your itinerary:

Taipei - Beijing - Hong Kong

Is perfectly eligible for 144 hour visa free transit. Of course, the flights between the mentioned places have to be nonstop.

Hong Kong, Taiwan and Macau, although not countries, but Chinese territories, are considered as countries for visa free transit purposes. So, if you use Taiwan for visa free transit purposes, this is the same as if you were using, lest's say, South Korea or Canada for the same thing.